Analytical and Experimental Study of Turbulent Flow Drag ... · The thesis starts with a...

Analytical and Experimental Study of Turbulent Flow Drag Reduction and Degradation

with Polymer Additives

By

© Xin Zhang

A Thesis submitted to the

School of Graduate Studies

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Faculty of Engineering and Applied Science

Memorial University of Newfoundland

May 2020

St. John's Newfoundland

i

Abstract

Flow friction reduction by polymers is widely applied in the oil and gas industry for flow

enhancement or to save pumping energy. The huge benefit of this technology has attracted many

researchers to investigate the phenomenon for 70 years, but its mechanism is still not clear. The

objective of this thesis is to investigate flow drag reduction with polymer additives, develop

predictive models for flow drag reduction and its degradation, and provide new insights into the

drag reduction and degradation mechanism.

The thesis starts with a semi-analytical solution for the drag reduction with polymer

additives in a turbulent pipe flow. Based on the FENE-P model, the solution assumes complete

laminarization and predicts the upper limitation of drag reduction in pipe flows. A new

predictive model for this upper limit is developed considering viscosity ratios and the

Weissenberg number - a dimensionless number related to the relaxation time of polymers. Next,

a flow loop is designed and built for the experimental study of pipe flow drag reduction by

polymers. Using a linear flexible polymer - polyethylene oxide (PEO) - in water, a series of

turbulent flow experiments are conducted. Based on Zimm’s theory and the experimental data, a

correlation is developed for the drag reduction prediction from the Weissenberg number and

polymer concentration in the flow. This correlation is thoroughly validated with data from the

experiments and previous studies as well.

To investigate the degradation of drag reduction with polymer additives, a rotational

turbulent flow is first studied with a double-gap rheometer. Based on Brostow’s assumption, i.e.,

the degradation rate of drag reduction is the same as that of the molecular weight decrease, a

correlation of the degradation of drag reduction is established, along with the proposal of a new

theory that the degradation is a first-order chemical reaction based on the polymer chain scission.

ii

Then, the accuracy of the Brostow’s assumption is examined, and extensive experimental data

indicate that it is not correct in many cases. The degradation of drag reduction with polymer

additives is further analyzed from a molecular perspective. It is found that the issue with

Brostow’s theory is mainly because it does not consider the existence of polymer aggregates in

the flow. Experimental results show that the molecular weight of the degraded polymer in the

dilute solution becomes lower and the molecular weight distribution becomes narrower. An

improved mechanism of drag reduction degradation considering polymer aggregate is proposed -

the turbulent flow causes the chain scission of the aggregate and the degraded aggregate loses its

drag-reducing ability. Finally, the mechanism of drag reduction and degradation is examined

from the chemical thermodynamics and kinetics. The drag reduction phenomenon by linear

flexible polymers is explained as a non-spontaneous irreversible flow-induced conformational-

phase-change process that incorporates both free polymers and aggregates. The entire non-

equilibrium process is due to the chain scission of polymers. This theory is shown to agree with

drag reduction experimental results from a macroscopic view and polymer behaviours from

microscopic views.

The experimental data, predictive models, and theories developed in this thesis provide

useful new insights into the design of flow drag reduction techniques and further research on this

important physical phenomenon.

iii

Acknowledgements

I would like to thank my supervisor, Prof. Xili Duan, for his guidance and help

throughout this thesis. At the beginning of this work, I once felt lost and did not know how to

solve the research problems, and even had difficulty in writing papers and communicating with

other people. Dr. Duan provided useful help in all these aspects. Without his help, I could not

complete my Ph.D. program. I also wish to thank my co-supervisor, Prof. Yuri Muzychka, for

his help in fluid mechanics theories and discussions to improve my research work. I am grateful

to Prof. Kevin Pope and Prof. Yan Zhang for their valuable suggestions on my research. Besides,

I also thank Prof. Anand Yethiraj from the department of physics and physical oceanography at

the Memorial University of Newfoundland for the use of rheometer, and Prof. Zongming Wang

from the department of chemical engineering at the China University of Petroleum (East China)

for helping me to establish the flow loop. I am also grateful to the financial support from the

Memorial University of Newfoundland and the NL Innovation Council. Last, I wish to thank my

parents who always support me for my Ph.D.

iv

Contents

Abstract ........................................................................................................................................... i

Acknowledgements ...................................................................................................................... iii

List of Figures .............................................................................................................................. vii

List of Tables ................................................................................................................................. x

List of Symbols ............................................................................................................................. xi

Chapter 1 Introduction of Drag Reduction by Polymers .......................................................... 1

1.1 Drag Reduction Research History ......................................................................................... 1

1.1.1 Top Three Achievements of Drag Reduction Research ................................................. 3

1.2 Drag Reduction Applications ................................................................................................ 4

1.3 Thesis Objective, Structure and Highlights .......................................................................... 8

Chapter 2 Literature Review ..................................................................................................... 11

2.1 Drag Reduction ................................................................................................................... 11

2.1.1 The Problem of Drag Reduction Mechanism ............................................................... 11

2.1.2 Modelling of Drag Reduction ....................................................................................... 13

2.1.3 Scale-Up Effect and Correlations ................................................................................. 14

2.1.4 Experimental Methods .................................................................................................. 19

2.2 Degradation of Drag Reduction .......................................................................................... 22

2.3 Mechanism of Drag Reduction by Polymers ...................................................................... 26

2.3.1 Introduction .................................................................................................................. 26

2.3.2 Unresolved Problems in the Proposed Drag Reduction Mechanism ............................ 26

2.4 Summary ............................................................................................................................. 29

v

Chapter 3 A Semi-Analytical Model for the Upper Limit of Drag Reduction with Polymer

Additives ...................................................................................................................................... 30

3.1 Model Formulation .............................................................................................................. 30

3.2 Model Tuning and Discussions ........................................................................................... 35

3.3 Summary ............................................................................................................................. 42

Chapter 4 Experimental Correlation for Pipe Flow Drag Reduction Using Relaxation Time

....................................................................................................................................................... 43

4.1 Theories and Correlation Formulation ................................................................................ 43

4.1.1 Clarification of the Relaxation Time, Deborah Number, and Weissenberg Number ... 43

4.1.2 Correlation Formulation ............................................................................................... 46

4.1.3 Concentration Range for Correlation ........................................................................... 48

4.2 Experimental Setup and Procedure ..................................................................................... 50

4.3 Experimental Results........................................................................................................... 54

4.3.1 Drag Reduction Experimental Data and Analysis ........................................................ 54

4.3.2 Correlation and Validation ........................................................................................... 60

4.4 Discussions .......................................................................................................................... 62

4.5 Summary ............................................................................................................................. 64

Chapter 5 Mechanism and Correlation for Degradation of Drag Reduction by Polymers in

Rotational Flows.......................................................................................................................... 65

5.1 Experiment .......................................................................................................................... 65

5.2 A New Theory of Degradation Mechanism by Polymers ................................................... 67

5.3 Results and Discussions ...................................................................................................... 70

5.4 Summary ............................................................................................................................. 75

vi

Chapter 6 A New Molecular View of Polymer Degradation in Drag-Reducing Flow.......... 77

6.1 Examination of Brostow’s Assumption .............................................................................. 77

6.2 Aggregate Degradation in Drag-Reducing Flow ................................................................ 81

6.3 Molecular Weight Distribution Shift in Drag-Reducing Flow............................................ 85

6.4 Summary ............................................................................................................................. 87

Chapter 7 Mechanism of Drag Reduction and Degradation from Chemical

Thermodynamics and Kinetics .................................................................................................. 88

7.1 Explanation of Drag Reduction by Polymers ...................................................................... 88

7.2 Discussions .......................................................................................................................... 93

7.3 Summary ............................................................................................................................. 98

Chapter 8 Conclusions and Future Work .............................................................................. 100

8.1 Conclusions ....................................................................................................................... 100

8.2 Future Work ...................................................................................................................... 101

8.2.1 Environmental-Friendly Drag-Reducing Agents........................................................ 101

8.2.2 Anti-Degradation ........................................................................................................ 102

8.2.3 Oil-Soluble Polymers and Multiphase Flow .............................................................. 102

8.2.4 Synergy of Drag Reduction by Polymers and Surface Modification ......................... 103

Reference ................................................................................................................................... 104

Appendix .................................................................................................................................... 133

Appendix 1 How to Obtain the Weissenberg Number from 𝝁𝑺, u, d, and N .......................... 133

Appendix 2. Why the Correlation Format of Eq. 4-8 Is Proposed .......................................... 135

Appendix 3. List of Publications ............................................................................................. 136

vii

List of Figures

Figure 1-1 Summary of publications regarding drag reduction by additives ................................. 2

Figure 1-2 Thesis structure ............................................................................................................. 8

Figure 2-1 A typical result of drag reduction by polymers (data from Savins (1967)) ................ 13

Figure 3-1 Comparison of experimental data with calculated drag reduction from the upper limit

model (data from Japper-Jaafar et al. (2009)) ............................................................................... 36

Figure 3-2 The linear relationship between 𝛼 and DRExp ............................................................. 38

Figure 3-3 Relationships between measured fluid properties in rheometer, calculated upper limit

of drag reduction (DRCal), and estimated drag reduction in pipe flow (DRExp) ............................ 39

Figure 3-4 Drag reduction performance of two DRAs ability at the same velocity originally from

Abubakar et al. (2014) and at the same concentration from Kamel & Shah (2009)..................... 41

Figure 4-1 Polymer behaviors in a dilute solution ........................................................................ 44

Figure 4-2 (a) Dilute polymer solution when c is less than C*, (b) Critical state when c is equal to

C*, (c) Semi-dilute solution when c is greater than C* ................................................................. 49

Figure 4-3 Schematic diagram (a) and photo (b) of the flow loop ............................................... 51

Figure 4-4 Benchmark test for the flow loop ................................................................................ 54

Figure 4-5 The relationship between friction factor and Reynolds number at different

concentrations. (a) When M = 106 g/mol, (b) When M = 2×106 g/mol, (c) When M = 4×106 g/mol

....................................................................................................................................................... 57

Figure 4-6 The relationship between drag reduction and Reynolds number at different


....................................................................................................................................................... 58

viii

Figure 4-7 Drag reduction at different Weissenberg numbers: (a) when M = 106 g/mol; (b) when

M = 2 × 106 g/mol; (c) when M = 4 × 106 g/mol .......................................................................... 60

Figure 4-8 Comparison of drag reduction data from experiments and predictions by the

developed correlation in Eq. 4-19 ................................................................................................. 61

Figure 5-1 (a) Geometry of a double gap (DG) rheometer, R1 = 11.909 mm, R2 = 12.328 mm, R3

= 13.332 mm, R4 = 13.797 mm; (b) photo of the rheometer (The copyright of this photo belongs

to the Anton-Paar website)............................................................................................................ 66

Figure 5-2 Torque of DI water and polymer solution (CP = 20 ppm, T = 318 k, 𝑟 = 8000 s-1, M =

2×106 g/mol) ................................................................................................................................. 71

Figure 5-3 Experiment data and correlation by Eq. 5-9 (CP = 35 ppm, T = 318 k, 𝑟 = 7000 s-1, M

= 4×106 g/mol) .............................................................................................................................. 72

Figure 5-4 Comparison between experimental data and prediction by Eq. 5-15 and Eq. 5-16 at

different fluid and flow conditions: (1) M = 4×106 g/mol, CP = 50 ppm, T = 298 k, 𝑟 = 8000 s-1,

ARE = 11.7%; (2) M = 2×106 g/mol, CP = 50 ppm, T = 338 k, 𝑟 = 6000 s-1, ARE = 6.43%; (3) M

= 106 g/mol, CP = 35 ppm, T = 298 k, 𝑟 = 7000 s-1, ARE = 6.27%; (4) M = 2×106 g/mol, CP = 35

ppm, T = 318 k, 𝑟 = 7000 s-1, ARE = 14.8% ................................................................................. 74

Figure 6-1 The relationship of DR(t)/DR(0) and M(t)/M(0): (a) revised from Lee et al. (2002) and

(b) from Vanapalli et al. (2005) .................................................................................................... 78

Figure 6-2 The relationship between drag reduction and radius of hydrodynamics from Van Dam

& Wegdam’s paper (1993) ............................................................................................................ 81

Figure 6-3 Possible aggregate degradation mechanism in drag-reducing flow ............................ 83

Figure 6-4 Continuous and batch degradation test of PEO (molecular weight 4 106 g/mol,

concentration 20 ppm, temperature 20 °C and shear rate 6500 s-1) .............................................. 84

ix

Figure 6-5 The molecular weight distribution before and after degradation ................................ 86

Figure 6-6 Molecular weight distribution of degraded polymer in drag reduction (data from

Vanapalli et al. (2005)) ................................................................................................................. 87

Figure 7-1 Polymer transition between the coiled state and the stretched state (data from Fidalgo

et al. (2017)) .................................................................................................................................. 89

Figure 7-2 The proposed explanation of the drag reduction and degradation by linear flexible

polymers ........................................................................................................................................ 93

Figure 7-3 Drag reduction oscillation versus distance or time (a: data from McComb & Rabie

(1978); b: data from Bewersdorff & Petersmann (1987); c: data from Camail et al. (1998); d:

data from Strelnikova & Yushchenko (2019)).............................................................................. 96

Figure 7-4 Field drag reduction data in an industrial pipeline and their results after Fourier series

transformation (data from Ref. by Strelnikova & Yushchenko (2019)) ....................................... 98

x

List of Tables

Table 1-1 Summary of application drag reduction by polymers in the industry ............................ 7

Table 2-1 Summary of drag reduction correlation based on the Reynolds number ..................... 20

Table 2-2 Summary of drag reduction correlation based on the Deborah number or Weissenberg

number .......................................................................................................................................... 23

Table 4-1 Overlap concentration (C*) at different molecular weights .......................................... 50

Table 4-2 Summary of previous experimental data and conditions for correlation validation .... 61

xi

List of Symbols

a Monomer length m

C* Overlap concentration ppm

CP Concentration of DRAs ppm

𝑐 Conformation tensor of the polymer -

d Diameter m

De Deborah number -

DR% Drag reduction -

Ea Activation energy J/mol

f Friction factor -

G Modulus Pa

G' Storage modulus Pa

G'' Loss modulus Pa

I Unit tensor -

k Chemical reaction rate constant s-1

k0 Pre-exponential factor s-1

l Maximum length of the polymer; length of the pipe m

M Average molecular weight g/mol

N Polymerization degree -

NA Avogadro constant mol-1

P Pressure drop along a certain direction Pa

Q Flow rate L/min

�⃗⃗� End-to-end vector of the polymer -

xii

RG Radius of gyration of polymer m

RH Radius of hydrodynamics of polymer m

�̇� Shear rate s-1

Re Reynolds number -

Reb Reynolds number based on bulk velocity -

t Time s

T Absolute temperature; torque k; N·m

�⃗⃗� Stress tensor of the polymers -

tP Observation time s

�⃗⃗� Velocity vector -

Ub Bulk velocity m·s-1

Wi Weissenberg number -

x Distance m

Greek symbols

α Ratio of drag reduction; order of chemical reaction -

β Ratio of viscosity -

δ Uncertainty -

Ɛ Roughness of pipe m

λ Relaxation time s

μ Viscosity Pa·s

ρ Density kg·m-3

1

Chapter 1 Introduction of Drag Reduction by Polymers

1.1 Drag Reduction Research History

Some significant discoveries are often out of expectations, and drag reduction by

polymers is one example. Toms found this remarkable phenomenon while he did not aim to

research the pressure drop in the middle of 1946 (Toms, 1948). He was studying the degradation

of polymer (polymethyl methacrylate) in a dilute solution (Chlorobenzene as the solvent) in a

turbulent pipe flow, and the pressure drop was measured by a mercury u-tube manometer.

Amazingly, Toms found that the pressure drop with polymers was lower than the one without

polymers. Then he wished to find an explanation about this phenomenon from published works

but failed since no work regarding it was released before. As a chemist, Toms did not understand

this phenomenon. After he found this phenomenon, Oldroyd, another principal researcher in drag

reduction by polymers, showed great interest in it. As a researcher in the Research Laboratory of

Courtaulds with a mathematical background, Oldroyd mentioned that this phenomenon might be

related to the modification of near-wall structure in turbulent flow (Toms, 1948 and 1977).

Before Toms observed the drag reduction phenomenon, Brautlecht & Sethi (1933) and

Forrest & Grierson (1931) found that in pumping paper pulps, adding dilute suspensions could

reduce the friction in the pipe. However, they used a figure to indicate this phenomenon but did

not point out that it was the drag reduction phenomenon induced by chemical additives. Thus,

the finding of the drag reduction phenomenon was still credited to Toms. In 1948, Toms

presented his findings in the 1st International Congress on Rheology in Scheveningen

Netherlands (Toms, 1948). However, this important finding did not receive much attention after

it was published for the first time (Toms, 1977).

2

Figure 1-1 is the summary of Nadolink & Haigh’s work (1995), who summarized the

publication number about drag reduction by additives. It includes journal papers, conference

papers, technical reports, book chapters, presentations and theses. In total, approximately 5000

papers were published from 1948 to 1995. Data of the publication number after 1995 are not

available.

Figure 1-1 Summary of publications regarding drag reduction by additives

Figure 1-1 indicates that, from 1948, when this phenomenon was published, to the 1960s,

before the oil crisis, this critical research area received little attention since few publications

regarding this topic were released. But from the 1960s, this phenomenon received remarkably

notice, and this was shown in the publication number because of the energy crisis in the late

1960s (oil embargo, 1967) and early 1970s (the first oil crisis, 1973) (Daoudi & Dajani, 1984;

Houthakker, 1983). These energy crises forced oil companies to develop technologies to save

3

budget in every aspect of the oil and gas industry, including transportation. In this case,

researchers received more funding from governments and companies in this area, and published

more papers since these two energy crisis.

1.1.1 Top Three Achievements of Drag Reduction Research

The top three achievements of drag reduction by polymers in the author’s view are the

following ones:

1970s’: Virk’s maximum drag reduction (Virk et al., 1970; Virk, 1975)

In the 1970s, Virk summarized the drag reduction data in several works and proposed the

famous Virk’s asymptote (maximum drag reduction line by polymers). This asymptote is

independent of polymer type, polymer concentration and viscosity etc. and only dependent on

the Reynolds number. After Virk released this result, researchers started to use this maximum

drag reduction line as a reference to show that all friction factors in drag-reducing flows should

be less than the one predicted by Virk’s asymptote which will be shown later. This result is also

the most widely-accepted conclusion in the drag reduction research. The paper in 1970 is cited

by more 300 times and the paper in 1975 is cited more than 1300 times.

1980s’: Application in Alaska (Burger et al., 1980 and 1982)

The Trans-Alaska Pipeline System (TAPS) is the most successful application of drag

reduction by polymers. In the 1970s, as mentioned above, the harsh situation for oil and gas

industry forced oil companies to use new technologies to save the budget and increase oil output.

In this condition, Alaska Pipeline Service Company first did several laboratory-scale

experiments in 2.54 and 5.04 cm diameter pipes. After these tests, they did more tests in larger

scale pipes (35.6 cm-diameter). They were looking for the relationship between the drag

reduction in a small diameter pipeline and the drag reduction in a large-scale application (122

4

cm-diameter). The detailed information can be found in the two references mentioned above.

This application is successful, but the empirical correlation and its method are not very scientific

since they only used empirical coefficients to establish the correlation without reasonable

interpretations.

1990s’: De Gennes’s Nobel Prize lecture about soft matter (De Gennes, 1992)

De Gennes (1932-2007) was a leading soft-matter scientist who won the Nobel Prize in

1991. In his Nobel-prize winning speech about soft matter, he also mentioned the drag reduction

by polymers. This presentation is the most famous work for the drag reduction by polymers as an

individual research topic. In this speech, he defined the concept of soft matter (complex fluid)

with two features, complexity and flexibility, which also works for drag reduction by polymers.

1.2 Drag Reduction Applications

How much energy or cost is saved by oil companies in polymer drag reduction

technology? Table 1.1 summarizes the drag reduction application in the oil-gas industry. On

average, the drag reduction is about 20% based on the drag reduction definition in Eq. 1-1 (∆𝑃𝑃

and ∆𝑃𝑆 are the pressure drop with and without polymers (pure solvents) under the same flow

rate).

𝐷𝑅% =∆𝑃𝑆 − ∆𝑃𝑃

∆𝑃𝑆× 100% (1-1)

If the drag reduction defined in Eq. 1-1 cannot represent how much energy is saved by

companies, there is an alternative method to show how much this effect can help companies to

reduce costs. The flow rate increase (FI%) is a function of drag reduction in Eq. 1-2 (Burger et

al., 1980 and 1982).

𝐹𝐼% =1

(1 − 𝐷𝑅%)0.55 − 1× 100% (1-2)

5

FI% describes the flow rate increase percentage if the pumping power remains the same.

From Table 1-1, it can be seen that the flow increase is significant, especially for the Alaska

pipeline, the most important drag reduction application. The flow increase is 560 m3/h

approximately from Eq. 1-2, more than 30,000,000 bbl/y (barrels oil per year). It means that this

drag reduction technology provides an “extra” 1.8 billion US dollar per year based on the current

oil price, i.e., approximately 60 US dollars per barrel. This technology offers a substantial

economic benefit to the oil gas company.

There is also a positive side-effect of using drag-reducing polymers – the anti-corrosion

effect in the transportation pipeline (Schmitt et al., 2001; Sedahmed et al., 1979, 1984 and 1999;

Zahran et al., 1997 and 1998). This means that once polymers are used in transportation, not only

pressure drop is decreased, but also the corrosion by the crude oil decreases. This side effect can

lead to fewer replacements of pipeline and increase oil transportation safety.

Besides the application in oil and gas industry, drag reduction by polymers can also be

applied in the other areas, i.e. irrigation (Khalil et al., 2002; Phukan et al., 2001), heating in

building (to reduce the heat loss) (Kotenko et al., 2019; Myska & Mik, 2003), firefighting

(Fabula et al., 1971; Figueredo et al., 2003), reservoir hydraulic fracturing (Ibrahim et al., 2018;

Nguyen et al., 2018; Shah et al., 2010; Nguyen et al., 2018), and municipal wastewater

transportation (Sellin, 1978 and 1983), even in drinking water transportation without affecting

drinking water quality (Edomwonyi-Out et al., 2018). This drag reduction effect can also be used

in blood pressure control. However, due to the potential toxicity of polymer to human, this

application is still in the lab stage, not in the human experiment (Coleman et al., 1987; Faruqui et

al., 1978; Hutchison et al., 1987; Kameneva et al., 2004; Polimeni et al., 1985 and 1989;

Sawchuk et al., 1999; Unthank et al., 1992).

6

Previous sections mentioned the application of drag reduction in internal flows. Even in

external flow, such as the shipbuilding industry, the phenomenon is still useful. White (1966)

first argued that the drag-reducing effect of polymers could reduce the resistance of submarine

vessel. Chahine et al. (1993) and Khomyakov & Elyukhina (2019) followed this view and used

polymers to reduce the friction between the propeller of ship and sea water. Not only frictions,

but also noise made by the propeller could be decreased by polymers (Oba et al., 1978; Reitzer et

al., 1976).

7

Table 1-1 Summary of application drag reduction by polymers in the industry

Note: oil price is 60 US dollar per barrel and drag reducing polymer price is 3 US dollar/kg based on the current price. Even if

the investment of polymer injection stations is 10 million US dollars (an estimation), there is still important benefit from the drag

reduction technology.

Reference Location Liquid Polymer name

(Commercial

name)

Concentration

(ppm)

Inner

diameter

(cm)

Length

(km)

Flow

rate

(m3/h)

Drag

reduction

(%)

Flow

increase

(%)

Flow

increase

(Million

bbl/year)

Saving

(Million

US

$/year)

Cost of

polymers

(Million US

dollar/year)

Burger et

al., 1980

and 1982

Alaska,

U.S.

Crude oil NA (CDR) 5-20 122 1287 7950 12 on

average

7 30 1830 0.17-0.68

Cao et al.,

2018

Shandong,

China

Crude oil Poly-α-

olefin

20-30 20 15 226 16-33 10 1.2 70 0.11-0.17

Carradine et

al., 1983

Montana,

U.S.

Gasoline NA (CDR) 14 20 96 207 28 20 1.9 130 0.076

Diesel 5 196 26 18 1.9 110 0.026

Lescarboura

et al., 1971

Oklahoma

, U.S.

Crude oil NA 230 21 46 276 18 11 1.6 100 1.66

250 31 52 559 17 11 3.4 200 3.67

Muth et al.,

1985

Conoco

Inc., U.S.

Gasoline NA (CDR) NA 20 97 199 28 24 2.6 150 NA

Fuel oil 15 80 20 14 2.3 90 NA

Muth et a.,

1986

Kansas.

U.S.

Crude oil NA (CDR) 38 41 262 713 23 15 5.9 350 0.71

Yang et al.,

2018

Sichuan,

China

Diesel Poly-α-

olefin

6-16 51 130 1232 28-55 20-55 13-37 820-

2240

0.19

8

1.3 Thesis Objective, Structure and Highlights

The objective of this thesis is to investigate flow drag reduction with polymer additives,

develop predictive models for flow drag reduction and its degradation, and provide new insights

into the drag reduction and degradation mechanism. The relationship between these different

aspects and the thesis structure is shown in Figure 1-2.

Figure 1-2 Thesis structure

9

Chapter 1 (background): a short history of drag reduction, applications and thesis

structure are repsented.

Chapter 2 (literature review) : a review of relavent literature for the reserch topic is

represented.

Chapter 3 (analytical research about the semi-analytical solution of drag reduction): from

governing equations for drag reduction by polymers and several assumptions, a semi-

analytical solution to predict the drag reduction is developed and it is validated by

experimental data.

Chapter 4 (experimental research about the drag reduction prediction using Zimm’s

relaxation theory): an experimental study in a flow loop is peformed to investigate the

drag reduction by polymers based on Zimm’s relaxation theory.

Chapter 5 (correlation and mechanism of drag reduction degradation): degradation of

drag reduction is studied to develop a correlation and a new degradation mechanism from

chemical reaction view is provided.

Chapter 6 (fundmental research about the correctness of the widely-accepted Brostow’s

assumption): experimental data are used to show the inaccuracy of Brostow’s theory. An

improvded degradation mechanism of drag reduction is then proposed.

Chapter 7 (proposed mechanism of drag reduction and degradation): chemical

thermodynamics and kinetics are introduced to propose a new drag reduction and

degradation mechanism by polymers.

Chapter 8 (conclusions and recommendation for future work): conclusions in previous

chapters are summarised and the future work recommendations are provided.

10

The novelty and highlight of this thesis are summarized as follows:

A semi-analytical solution is proposed from the FENE-P model and validated by

experimental results.

The definition of the relaxation time and Weissenberg number are clarified, and a

correlation of drag reduction prediction is provided from the polymer relaxation.

The degradation of drag reduction is explained as a first-order chemical reaction, and this

view is supported by experimental data.

A revised mechanism of degradation of drag reduction is provided, which incorporates

the degradation of free polymer and degradation of polymer aggregate.

The mechanism for flow drag reduction and degradation is proposed based on the

observation of molecular behavior in the microscope and validated by the experimental

data.

11

Chapter 2 Literature Review

2.1 Drag Reduction

2.1.1 The Problem of Drag Reduction Mechanism

For every research topic, one of the fundamental problems is to understand the

mechanism of the phenomenon. On the topic of drag reduction by polymers, researchers also

hope to find the mechanism to completely understand this phenomenon. However, such a

mechanism is still not available even this has been investigated for more than 70 years. The

detailed reasons will be shown in the following section.

On the application side, the aim of the drag reduction research has been to help the design

of oil transportation. Most studies use a flow loop to simulate this process. In the flow loop

experiment, the velocity (flow rate), concentration of polymers, polymer type, molecular weight,

pipe diameter and temperature can be changed to investigate how these parameters affect the

drag reduction. In the drag reduction study, friction factor is often used to manifest the resistance

in drag-reducing flow. Friction factor, f, is defined in Eq. 2-1. In this equation, d is the pipe

diameter; ΔP is the pressure drop; l is the pipe length; ρ is the fluid density; Ub is the bulk

velocity.

𝑓 =𝑑∆𝑃

2𝑙𝜌𝑈𝑏2 (2-1)

Figure 2-1 shows a typical result of drag reduction research by drag-reducing agents, also

known as the most acknowledged result, polymer or surfactant in a pipe or channel flow. The

friction factor in the drag-reducing flow should be enveloped by several lines: friction of laminar

flow and its extension (Eq. 2-2, Re ≤ 2300 in pipe or channel flows), friction factor of fully

turbulent flow by Blasius equation (Eq. 2-3, 4000 ≤ Re ≤ 105 in pipe or channel flows), friction

12

factor by Virks’s asymptote (maximum drag reduction line by polymers, first introduced by Virk

et al. in 1970, Eq. 2-4, 4000 ≤ Re ≤ 4×104 in pipe or channel flows) and friction factor by

Zakin’s asymptote (maximum drag reduction line by surfactants, first introduced by Zakin et al.

in 1996, Eq. 2-5, 4000 ≤ Re ≤ 4×104 in pipe or channel flow). Note: the drag reduction by

surfactants in this thesis is not investigated, but it is to show that the potential of drag reduction

by surfactants may be higher than polymers even though concentration for surfactants in the

drag-reducing flow is higher than the one of polymers, which may increase the cost of drag

reduction. The drag reduction started to happen from the onset point of turbulent flow (the

Reynolds number 4000 for pipe and channel flow). With a higher Reynolds number, the friction

factor will be decreased further until the optimum Reynolds number appears under a given

concentration. Under the optimum Reynolds number, the drag reduction reaches its maximum

value. If the Reynolds number is further increased, the drag reduction will be smaller, which is

explained in our work (Zhang et al., 2018), and also will be shown in Chapter 3.

𝑓 =16

𝑅𝑒 (2-2)

𝑓 = 0.079𝑅𝑒−0.25 (2-3)

𝑓 = 0.59𝑅𝑒−0.58 (2-4)

𝑓 = 0.32𝑅𝑒−0.55 (2-5)

13

Figure 2-1 A typical result of drag reduction by polymers (data from Savins (1967))

2.1.2 Modelling of Drag Reduction

From the 1970s, researchers started to modify the Reynolds stress model (Hassid, 1979),

turbulent energy dissipation (ɛ-k) model (Hassid & Poreh, 1975 and 1978) to investigate the drag

reduction by polymers. Others used similar models, focusing on the modified boundary layer by

polymers (Anderson & Wu, 1971; Test, 1974). One of the main results in these studies is the

velocity profile in the drag-reducing turbulent flow, validated by several experimental results

(Eskin 2017; Yang 2009; Yang & Ding, 2013 and 2014; Yang & Dou, 2008), where the velocity

profile data were measured by PIV (particle imaging velocimetry).

Nowadays, these models are no longer used. In most papers, only one model is used, i.e.,

the FENE-P model (Finitely Extensible Nonlinear Elastic and P for the Peterlin equation). In this

model, the traditional Naiver-Stokes equation is modified with consideration of the effects of

polymers in the turbulent flow. The drag-reducing polymer is regarded as a dumbbell. The key in

14

this model is the relaxation time of polymers in the flow, which will be shown later. Several

studies tried to use experimental data to verify this model. Den Toonder (1997) and Ptasinski et

al. (2003) used the velocity profile measured by PIV to validate the velocity profile in the

numerical simulation. The model can be validated by some experimental results, but this model

cannot predict all the drag reduction data. Researchers still need empirical correlations to predict

the drag reduction. In the following section, a summary of empirical drag reduction prediction

equations is shown.

2.1.3 Scale-Up Effect and Correlations

As mentioned above, researchers cannot fully understand the drag reduction phenomenon

and further propose a mechanism which is accepted universally, so the focus has been on

predicting the drag reduction in experimental conditions. In the prediction of drag reduction by

polymers, the most critical problem is the scale-up effect: most drag reduction studies are

performed in small diameter pipes, i.e. the lab scale. However, the diameter in the application of

oil and gas transportation is much larger than the lab-scale one. Thus, it is necessary to use some

methods to predict the drag reduction in the large scale from data obtained in a small scale.

Researchers have introduced several criteria for this scale-up effect.

Hoyt in his series of works (Hoyt, 1991; Hoyt and Sellin, 1993) proposed a similarity law

to predict the friction factor in the drag-reducing flow in Eq. 2-6. From this similarity law, drag

reduction data from a small diameter pipe can be used to predict the drag reduction in a large

diameter pipe.

𝑅𝑒2 = 𝑅𝑒1√𝑓2

𝑓1

𝑑2

𝑑1 (2-6)

If this equation is rearranged, Eq. 2-7 can be obtained.

15

𝑅𝑒1√𝑓1

𝑑1=

𝑅𝑒2√𝑓2

𝑑2 (2-7)

From this equation, the 𝑅𝑒√𝑓 𝑑⁄ can be treated as a constant. The Reynolds number is a

function of pipe diameter, bulk velocity, fluid density and viscosity. So, it can be further

rewritten as:

𝑅𝑒√𝑓

𝑑= 𝐶𝑜𝑛𝑠𝑡 (2-8)

From this equation, if the ratio of Re/d in the drag-reducing flow is the same, the friction

factor can be the same. But this criterion is not widely adopted since only limited data (Hoyt,

1991; Hoyt and Sellin, 1993) could validate it.

The previous similarity law only considered fluid properties (Reynolds number and

pressure drop) but did not consider the drag-reducing polymers since no parameters related to

polymers are involved in them. Instead, Virk & Baher (1970) proposed another similarity law

that combined the friction velocity and properties of the polymer solution, shown in Eq. 2-9. 𝜇𝑆

is the viscosity of the solvent and RG is the radius of gyration of polymers in the coiled state. 𝑢𝜏

is the friction velocity, a function of shear stress at the wall, defined in Eq. 2-10. Shear stress is a

function of friction factor, defined in Eq. 2-11.

𝑢𝜏

𝜇𝑆𝑅𝐺 = 𝐶𝑜𝑛𝑠𝑡 (2-9)

𝑢𝜏 = √𝜏𝑊

𝜌

(2-10)

𝜏𝑊 = 𝑓

𝜌𝑈𝑏2

2

(2-11)

Although no other work followed this criteria, it is still valuable since it points out that

the configuration of polymers - the gyration of radius of polymers, RG, is involved in drag

16

reduction. This important parameter of polymer inspires future work, which considers the

relaxation time, another critical parameter in drag reduction.

Relaxation process refers to the polymer’s behavior from the stretched state to the coiled

state, and the relaxation time describes the time from the stretched state back to the coiled state.

This property is very critical in drag reduction since it is the key to understand the mechanism of

drag reduction suggested by De Gennes (1986). The mechanism suspension will be discussed in

the following section.

The definition of relaxation time of polymer, 𝜆, is clear and it is related to two

dimensionless numbers related to correlation to predict the drag reduction, Deborah number (De)

and Weissenberg number (Wi), defined below (Dealy, 2010; Poole, 2012). tP is the residence

time of polymers in the flow and �̇� is the shear rate at the wall.

𝐷𝑒 =

𝜆

𝑡𝑃=

𝜆

𝑙𝑈𝑏

(2-12)

𝑊𝑖 = 𝜆�̇� = 𝜆8𝑈𝑏

𝑑 (2-13)

The definition of these two dimensionless variables has been discussed for many years in

many articles. Roriguez et al. (1969) introduced a “dimensional Deborah number” (not

dimensionless one) to correlate the drag reduction and this “dimensional Deborah number”, and

detailed information of this dimensional Deborah number can be found in this work. Different

from this method, the dimensionless Deborah number in several works (Darby & Chang, 1984;

Darby & Pivsa‐Art, 1991; Gordon, 1970; Sever & Metzner, 1967) was a function of the

relaxation time, shear rate, Reynolds number and many other fluid property variables, which

definition did not follow the original one. Since the Deborah number is critical to predict the

drag reduction, Gordon (1971) compared several definitions, but he failed to provide a

17

conclusive definition. Recently, this confusing dimensionless number is clarified in one paper

(Zhang et al., 2019): the diameter of the pipe is related to the Weissenberg number and the length

of the pipe is related to the Deborah number.

Several similarity laws exist, but none of them can be applied in all drag reduction

studies. Thus, most studies still used experimental data to establish many empirical correlations

to predict drag reduction. There are two types of empirical correlations, dimensional and

dimensionless.

Many empirical correlation methods, such as artificial intelligence approach and response

surface methodology, were used for the prediction of drag reduction (Karami et al., 2016; Zabihi

et al., 2019). Correlations by these methods have a good agreement between the prediction and

experimental data, but these methods are not very useful since these two methods can be

regarded as a complete black box, and are not helpful to manifest the mechanism. Besides, many

dimensional variables are involved in these correlations. As known to all, dimensionless numbers

are preferred in the classic fluid dynamics theory, so correlations by these methods are not very

good even though they can predict the data well. Similar drawbacks can also be found in other

works, which also use dimensional variables (Kamel and Shah, 2009; Zhao et al., 2018).

As shown above, since there are two types of scale-up effect criteria, there are also two

types of correlations, one based on the Reynolds number, and the other based on the

Weissenberg or Deborah number.

Dodge & Metzner (1959) first proposed a friction factor in the drag-reducing fluid. In this

equation, the generalised Reynolds number was employed, and two coefficients were functions

of the flow behaviour index, n’. Savins (1964) further expanded this equation. He argued that

two coefficients were more complicated than what Dodge & Metzner (1959) expected: these two

18

coefficients were functions, but they could not be represented in a simple method as Dodge &

Metzner (1959) suggested. In fact, they were dependent on the polymer type and concentration.

Some researchers used another type of Reynolds number in the correlation. Shah et al.

(2002) introduced another correlation based on the solvent-based Reynolds number to predict the

friction factor in the drag-reducing flow (detailed information of two coefficients were not

listed). Following this idea, Shah & Kamel (2010) and Shah and Vyas (2011) used two

polynomials to predict the drag reduction or friction factor in the drag-reducing fluids. Similar to

Savins’s (1964) conclusion, they also agreed that the concentration was involved in the drag

reduction prediction correlation. The Reynolds number in these correlations were based on the

solvent, ReS, not generalized ReG based on the polymer solution. A summary of these

correlations and their application range is shown in Table 2-1.

These models could only explain the data in their own work, and could not be validated

by others. Thus, researchers started to use another method to develop correlations based on the

Weissenberg number or Deborah number, Wi or De, whose idea may be from the scale-up effect

view based on the relaxation process of the polymer as mentioned above.

Owolabi et al. (2017) provided their different view on the relaxation time. They used the

CaBER (Capillary Breakup Extensional Rheometer) to measure the relaxation time, not from the

rheology equation, as shown by Owolabi et al. (2017). The model was applied for their data in a

semi-dilute polymer solution. The definition of their Weissenberg number is shown in Eq. 2-13.

This thesis will use another method, i.e., the Zimm’s theory, to estimate the relaxation time for

the dilute polymer solution (see Chapter 4). The new correlation from this method can predict

data in our experiments and other studies. Thus, there are three methods to obtain the relaxation

19

time of polymers in the solution: rheology equation, CaBER measurement and Zimm’s theory.

The summary of these equations is shown in Table 2-2.

2.1.4 Experimental Methods

Most drag reduction experiments are performed in a flow loop aiming to simulate the

industrial applications. Some are in a rotating disk apparatus or similar devices (Choi et al.,

1999; Kim et al., 1997 and 2002; Yang et al., 1994), whose importance is less than the one of the

flow loop experiment since this type of experiment cannot simulate the industrial application,

thus the publication number is also limited. In flow-loop experiments, there are two methods in

the drag reduction experiment: homogeneous and heterogeneous method (Bewersdorff et al.,

1993; Frings, 1988; Zhang et al., 2019). The homogeneous method means that polymers are

premixed with solvents and this polymer solution under a given concentration is transported into

the pipeline by a pump, centrifugal pump (non-positive displacement pump) in most cases. The

heterogeneous method means that polymers are not premixed in solvents; instead, solvents are

transported by a centrifugal pump and polymer solutions are injected by a positive displacement

pump, diaphragm pump in most cases. Unlike the centrifugal pump, diaphragm pump does not

have a rotational blade that causes polymer degradation. Instead, in a diaphragm pump, by a

reciprocating diaphragm pushing liquid, concentrated polymer solution in drag reduction study is

injected into the pipeline, so this pump does not cause the degradation of polymers. The

degradation of polymers is shown in the next section.

20

Table 2-1 Summary of drag reduction correlation based on the Reynolds number

Author Year Correlation Reynolds number Note

Dodge

and

Metzner

1959 1

√𝑓=

4

(𝑛′)0.75log (𝑅𝑒𝐺𝑓1−

𝑛′

2 ) −1.2

(𝑛′)1.2

Generalized Reynolds

number for power-law

fluids: 𝑅𝑒𝐺 =

𝑑𝑛′𝜌𝑈𝑏

2−𝑛′

𝐾′8𝑛′−1(3𝑛′+1

4𝑛′ )𝑛′−1

Power-law fluids:

𝜏𝑤 = 𝐾′ (8𝑈𝑏

𝑑)

𝑛′

Polymer: Polyacrylic acid

Concentration: 2000-5000 ppm

Reynolds number: 5000-36000

Savins 1964 1

√𝑓= 𝐴(𝑛′) log (𝑅𝑒𝐺𝑓1−

𝑛′

2 ) − 𝐵(𝑛′)

Polymer: Cellulose, vinyl

polymer and gum



Shah et

al. 2002 𝑓𝑃 = 𝐴 +

𝐵

𝑅𝑒𝑆𝐶

Reynolds number

based-on

solvent: 𝑅𝑒𝑆 =𝑑𝜌𝑈𝑏

𝜇𝑆

Polymer: PHPA and XCD



Three coefficients, A, B and C

are not shown in the original

paper.

Shah and

Kamel 2010

𝑓𝑃 = 6.19 × 10−3 + 9.68 × 10−22𝑅𝑒3

+ 3.88ln (𝑅𝑒𝑆)

𝑅𝑒𝑆

Polymer: ASP 700 and ASP 820


Reynolds number: 22000-

150000

Shah and

Vyas 2011 𝐷𝑅 = 𝐴 +

𝐵

ln 𝑅𝑒𝑆+

𝐶

(ln 𝑅𝑒𝑆)2+

𝐷

(ln 𝑅𝑒𝑆)3

For coiled pipe:

𝐴 = −1740𝐶𝑃 − 216

𝐵 = 54743𝐶𝑃 + 7413

𝐶 = −558708𝐶𝑃 − 84342

𝐷 = 1830200𝐶𝑃 + 320414 Polymer: ASP 700 and ASP 820


21


130000

For straight pipe:

𝐴 = 1436𝐶𝑃 − 174

𝐵 = −52475𝐶𝑃 + 5968

𝐶 = 640160𝐶𝑃 − 67898

𝐷 = −2610021𝐶𝑃 + 257873

Polymer: ASP 700 and ASP 820



130000

Note: PHPA, ASP-700 and ASP-800 for partially hydrolyzed polyacrylamide and XCD for polysaccharide gum

22

2.2 Degradation of Drag Reduction

When polymers experience turbulent flow for a long-time, the polymer chain scissions

happen by shear and the molecular weight decreases. This causes the degradation of drag

reduction by polymers in the drag-reducing flow. Polymers with a low molecular weight have a

lower efficiency of drag reduction ability and the drag reduction ability decreases with time.

There are two types of degradation studies, in rotational flow or pipe flow. In rotational flow, the

polymer solution is added in a rotating disk apparatus or a similar device (Choi et al., 2000; Dai

et al., 2018; Kim et al., 2000). It seems that this type of degradation is not related to the

degradation in the pipe flow since the geometries of rotational flow and pipe flow are completely

different. So, researchers also use pipe flow to test the degradation of drag reduction by

polymers, which aims to simulate the degradation in real oil and gas pipelines (Motta et al.,

2019; Soares et al., 2019). However, these studies did not achieve the goal. Because most

experiments are conducted in a closed flow loop, meaning that the polymers experience the

degradation caused by the centrifugal pump as mentioned before. The degradation happens at

two sections, in the pump and pipe. Thus, these two degradations cannot be differentiated.

Overall, it is argued these two methods are essentially the same, indicating that the degradation

test in the flow loop cannot simulate the degradation in long-distance pipe flow in the industry. If

anyone wants to do the degradation test and hopes to use these data for industrial usages, the

researcher must have a long-enough open flow loop (long-distance non-closed flow line to avoid

repetitive degradation by centrifugal pumps). One work (Jouenne et al., 2015) almost fits these

requirements. However, the pipeline is not straight, and several corners are involved, thus it is

not helpful to predict the drag reduction in straight pipelines.

23

Table 2-2 Summary of drag reduction correlation based on the Deborah number or Weissenberg number

Author Year Correlation Weissenberg or Deborah number Note

Darby

and

Chang

1984

𝑓𝑃 =𝑓𝑆

√1 + 𝐷𝑒2

𝐷𝑒 =0.0166(8𝑢𝜆 𝑑⁄ )𝑅𝑒𝑆

0.375(𝜇𝑆 𝜇0⁄ )0.5

[1 + (8𝑢𝜆 𝑑⁄ )2]0.25

Polymer: AP-30

Concentration: 100-

500 ppm

Reynolds number:

4000-100000

Method for obtaining

relaxation time and 𝜇0:

Jeffrey model

Darby,

and

Pivsa‐

Art

1991

𝐷𝑒 =0.0163(8𝑢𝜆 𝑑⁄ )𝑅𝑒𝑆

0.338(𝜇𝑆 𝜇0⁄ )0.5

[1 𝑅𝑒𝑆0.75⁄ + 0.00476(8𝑢𝜆 𝑑⁄ )2(𝜇𝑆 𝜇0⁄ )0.75]0.318

Polymer: PEO, PAM

E198, Rhodopol 23

(Xantham Gum)

Concentration: 100-

500 ppm

Reynolds number:

10000-100000



Jeffrey model

Gallego,

and

Shah

2009

𝐷𝑒 =1.186(𝑓𝑆𝑅𝑒𝑆)0.322(8𝑢𝜆 𝑑⁄ )

[1 + (8𝑢𝜆 𝑑⁄ )2]0.703(

𝜌𝑃𝜇𝑆

𝜌𝑆𝜇0)

0.192

For straight tubing

Polymer: ASP-700 and

ASP-820

Concentration: 300-

1000 ppm

Reynolds number:

12000-100000



Jeffrey model

24

𝐷𝑒 =1.67 × 10−3(𝑓𝑃𝑅𝑒𝑆)1.408(8𝑢𝜆 𝑑⁄ )

[1 + 1.097(8𝑓𝑃𝑅𝑒𝑆𝑢𝜆 𝑑⁄ )2]1.423(

𝜌𝑃𝜇𝑆

𝜌𝑆𝜇0)

0.113

For coiled tubing

Polymer: ASP-700 and

ASP-820

Concentration: 300-

1000 ppm

Reynolds number:

12000-100000



Jeffrey model

Owolabi

et al.

2017

𝐷𝑅% = 128 (1

1 + 𝑒0.5−𝑊𝑖

− 0.5) 𝑊𝑖 =

8𝑢

𝑑

Polymer: PAA

Concentration: 150-

200 ppm

Reynolds number:

6000-10000


relaxation time:

CaBER

Note: AP-30 for partially hydrolyzed polyacrylamide; PAM E198 for Polyacrylamide; PAA for Polycarylamide; Rhodopol 23 for

Xantham Gum; PAA for Polyacrylic acid; PEO for Polyethylene oxide.

25

There are two types of widely accepted correlations for the degradation of drag reduction

by polymers.

𝐷𝑅(𝑡) = 𝐷𝑅(0)𝑒−𝑘𝑡 (2-14)

𝐷𝑅(𝑡)

𝐷𝑅(0)=

1

1 + 𝑊(1 − 𝑒−ℎ𝑡)

(2-15)

Eq. 2-15 is developed from an assumption, Eq. 2-16.

𝐷𝑅(𝑡)

𝐷𝑅(0)=

𝑀(𝑡)

𝑀(0) (2-16)

In these equations, DR(0) is the drag reduction value at the initial state, and DR(t) is the

degradation value at any time during the degradation process; M(0) is the average molecular

weight at the initial state, and M(t) is the average molecular weight at any time during the

degradation process. W, h and k are constants. Both correlations can be used for prediction of

degradation of drag reduction by polymers in many works in both rotational flow and pipe flow

(Brostow et al., 2007; Choi et al., 2000; Hong et al., 2008; Le Brun et al., 2016). However, a

problem in these two equations is that the physical meaning of parameters is not clear. In this

thesis, Eq. 2-14 to Eq. 2-16 will be combined to develop a correlation to predict the drag

reduction degradation. The degradation of drag reduction is regarded as a first-order chemical

reaction since the polymer chain scission is involved in the degradation process (the original long

chain polymers disappear and new shot chain polymers form). Further analysis with

experimental data will show that Eq. 2-16 is not correct, because this assumption only considers

the single-distribution polymer (DNA as an example) and neglects the possible aggregate of

polymer in the turbulent drag-reducing flow.

26

2.3 Mechanism of Drag Reduction by Polymers

2.3.1 Introduction

Drag reduction by linear polymers and its degradation in turbulent flows have been

investigated for many years, but the exact mechanism of these processes is still not clear. Some

existing theories are summarized as follows. Lumley (1973) proposed a mechanism from a

viscosity view: polymers enter the boundary layer of turbulent flow and increase the viscosity in

this region, which damps the small eddies so that less turbulent energy is dissipated, and pressure

drop decreases. De Gennes (1986) suggested a mechanism from the elastic view: polymers in

turbulent flow experience extension; elasticity in this extension transition damps eddies in the

turbulent flow, so the pressure drop decreases. Brostow et al. (1999) provided a mechanism from

flow domain theory: polymers form flow domains surrounding the solvents and suppressing

eddies formation; eddies disappear and drag reduction happens. Camail et al. (2009) offered a

mechanism from polymer stretching: drag reduction happens at two consecutive stretching steps,

from aggregates to isolated coils and from isolated coils to stretched coils. A common problem

with these proposed mechanisms is that aggregates of polymers are neglected. Besides, the

critical energy introduced by Camail et al. (2009) was not very clear.

2.3.2 Unresolved Problems in the Proposed Drag Reduction Mechanism

There are several proposed mechanisms of drag reduction by polymers. However, most

of these are not validated with experimental data. Currently, the most promising mechanism is

the elasticity theory by De Gennes. He thinks that the polymer phase change process (from the

coiled state to the stretched state) could absorb the turbulent energy sot that more energy could

be used in the main direction of pipe flow and the conformation of polymers remains in the

stretched state (De Gennes, 1986; Tabor & De Gennes, 1986). But he did not observe this stretch

27

process then, because the first direct observation of polymer in a flow was done in 1992 (Smith

et al., 1992). Does the observation validate this theory? No. Several independent works reported

that the conformation of polymers (real-time length of polymers) observed by fluorescent

microscope was time-dependent, not a constant after stretching (Bakajin et al., 1998; Lueth &

Shaqfeh, 2009; Sachdev et al., 2016; Teixeira et al., 2005).

The other factor that makes the drag reduction hard to understand is the “molecular

individualism”, first introduced by De Gennes (1997). He claimed that the behaviour of a single

Macromolecule (DNA as an example) in flow was hard to predict – the macromolecule could

stretch, relax and even tumble, the last behavior indicating that polymers may have a rigid

feature even thought it is flexible. Current models cannot handle all these features. Thus,

researchers cannot have a good understanding of this phenomenon.

The huge difference between the simulation and experimental conditions also causes a

difficulty in understanding the mechanism. Experimental data were used to validate the FENE-P

model and a semi-empirical correlation to predict the drag reduction was proposed, but there is

still a concern about the gap between experiment and simulation: the Weissenberg number, Wi,

in these two types of study is very different. In most numerical studies, the order of magnitude of

the Weissenberg number is O(102), and Wi between 10 and 100 (Kim et al., 2007; Li & Graham,

2007; Tesauro et al., 2007; Thais et al., 2010; Zhu et a., 2018); while in experimental studies, the

order of magnitude will be O(101), usually less than 10 (Owolabi et al., 2017). This huge gap

causes the concern that maybe a new model should be developed, instead of the FENE-F model,

for the numerical simulation study.

The existence of aggregate in the dilute polymer solution is still in debate (Devanand &

Selser, 1990). If aggregates are also involved, the drag reduction by polymers will be more

28

complicated: the relaxation time of free polymers and aggregates are different. So, one should

not use just one parameter to consider the effects of a polymer with a given molecular weight

and molecular weight distribution.

The molecular weight distribution of drag-reducing polymers should be taken into

considerations (Hunston, 1974; Little & Ting, 1976). In most cases, drag-reducing polymers are

synthetic, so polymers have a wide distribution of molecule weight. Either the number-averaged,

weight-averaged or viscosity-averaged molecular weight is used to describe the overall property

of these drag-reducing polymers. But can these average molecular weights represent the true

property of the polymer in drag reduction analysis? No. A profile of molecular weight exists, and

the same molecular weight could represent totally different polymers. Let’s take the weight-

averaged molecular weight (most used in the literature) as an example. Imagine that there are

two polymers with the same type and weight-averaged molecular weight, while their molecular

weight distribution is different, one being narrow and the other broad. These two polymers have

different drag reduction abilities under the same flow condition. It is well known that there is a

critical molecular weight, and polymers that have a molecular weight less than the critical one

have no drag reduction ability in a certain condition. The polymer with a narrow molecular

weight distribution has more “useful” polymers while the polymer with a board profile has less.

More useful polymers can contribute more to the relaxation time, and with a better drag

reduction ability. Thus, it is not appropriate to use one single molecular weight to represent the

overall property of the drag-reducing polymer.

The method of preparing a polymer solution is another factor that may cause different

results in the drag reduction study. Any difference in the preparation method may cause the

difference of the polymer solution, i.e. mechanical or magnetic mixing, low or high temperature,

29

low or high rotation speed (shear rate which may cause the degradation discussed above), long or

short mixing time, type of mixing blade, etc. All these differences could affect the polymer

solution. Rowin et al. (2018) provided several clues about the optimal preparation methods, but

few studies about this specific question in the drag reduction are available.

2.4 Summary

This chapter presents a literature review on flow drag reduction and degradation. Some

general conclusions, models, and correlations, and the scale-up criteria are summarized. Several

proposed drag reduction and degradation mechanisms are discussed, along with the reasons why

there is no universally accepted mechanism in the literature. In the next chapter, a semi-

analytical solution of the drag reduction using the FENE-P model for drag-reducing flow will be

provided to show the upper limit of drag reduction with polymer additives.

30

Chapter 3 A Semi-Analytical Model for the Upper Limit of Drag

Reduction with Polymer Additives

In Chapter 3, a semi-analytical solution for drag reduction prediction is proposed based

on the FENE-P model. This solution is tuned by previous experimental results and able to predict

the upper limit of drag reduction with polymer additives. The main content of this chapter has

been published as a journal paper (Zhang, X., Duan, X., & Muzychka, Y. (2018). “Analytical

upper limit of drag reduction with polymer additives in turbulent pipe flow”. Journal of Fluids

Engineering, 140(5), 051204). The author of this thesis is the first author of this paper. The first

author developed the model, analyzed the data, prepared the manuscript, and made the revisions.

Prof. Duan and Prof. Muzychka as the second and third authors provided their suggestions on the

model development and helped in the revision of the manuscript.

3.1 Model Formulation

In this study, the classic FENE-P model is used to obtain a stress tensor �⃗⃗�, shown in Eq.

3-1, induced by polymer drag reduction agents (DRAs). This is the basis for drag reduction

analysis of polymeric flows and this stress tensor is the key property of the polymer related to

drag reduction. Zhang et al. (2013) summarized all equations of the FENE-P model so only a

brief review will be provided here. Also note that all the equations in this model are

dimensionless.

�⃗⃗� =1

𝑊𝑖[

𝑐

1 −𝑡𝑟𝑎𝑐𝑒(𝑐)

𝐿2

− 𝐼] (3-1)

In Eq. 3-1, 𝑐 is the conformation tensor of the polymers, defined as 𝑐 = ⟨�⃗⃗��⃗⃗�⟩, where �⃗⃗� is

the end-to-end vector of the polymer and the bracket indicates the average in the flow field. L is

31

the dimensionless maximum length of the polymer in the solution, estimated in the simulation

process; 𝐼 is a unit tensor; Wi is the Weissenberg number, defined as:

𝑊𝑖 = 𝜆�̇� (3-2)

where 𝜆 is the relaxation time of the drag-reducing agent and �̇� is the shear rate in the turbulent

flow.

The governing equation for drag reduction is a modified Naiver-Stokes equation as

follows:

𝜕�⃗⃗�

𝜕𝑡+ �⃗⃗� ∙ ∇�⃗⃗� = −∇�⃗� +

𝛽

𝑅𝑒𝑏∇2�⃗⃗� +

1 − 𝛽

𝑅𝑒𝑏∇ ∙ �⃗⃗� (3-3)

where �⃗⃗� is the dimensionless velocity vector of the turbulent flow with the bulk velocity 𝑈𝑏 as a

reference; ∇�⃗� is the dimensionless pressure gradient with a reference pressure, 𝜌𝑈𝑏2; length is

nondimensionalized by the diameter of the pipeline, d; time is nondimensionalized by a

characteristic time, 𝑑 𝑈𝑏⁄ ; 𝛽 is the ratio of the viscosity of the solvent to the viscosity of liquid

after the polymer is added at the zero shear rate, defined as follows:

𝛽 =𝜇𝑆

𝜇𝑆 + 𝜇𝑃,0 (3-4)

where 𝜇𝑆 and 𝜇𝑃,0 represent the viscosity of solvent and polymers added in at zero shear rate.

For tensor 𝑐, the governing equation is as follows:

𝜕𝑐

𝜕𝑡+ (�⃗⃗� ∙ ∇)𝑐 = 𝑐 ∙ (∇�⃗⃗�) + (∇�⃗⃗�)𝑇 ∙ 𝑐 − �⃗⃗� (3-5)

The governing equations are analyzed with the following assumptions and

simplifications.

Assumption 1: Previous publications have proved that DRAs can reduce the fluctuation

velocity or Reynolds stress (Amarouchene & Kellay, 2002; Iwamoto et al., 2005; Samanta et al.,

32

2009; Warholic et al., 2001). If all the fluctuation velocities disappear, and with a constant flow

rate (basic condition for drag reduction studies), a steady flow is achieved, with 𝜕�⃗⃗� 𝜕𝑡⁄ = 0, and

consequently 𝜕𝑐 𝜕𝑡⁄ = 0. If all turbulent structures disappear, turbulent flow becomes laminar

flow, which is supported by Kostic (1994), who found that there was a laminarization trend of

turbulent flow after DRA was added. Furthermore, this assumption has been partially proved by

experiments. Many results (Bizhani et al., 2015; Li et al., 2005; Paschkewitz et al., 2005; Shao et

al., 2002; Tamano & Itoh, 2011;) were published with PIV data to show that fluctuating velocity

(Reynolds stress) decreases when DRAs are added. The extreme situation is that all the turbulent

structures disappear, and the upper limit of drag reduction is achieved.

Assumption 2: uy is negligible as ux is more important in pipelines, which was also

assumed by Barenblatt (2008). The turbulent flow in pipelines is assumed to be fully developed,

therefore 𝜕𝑢𝑥 𝜕𝑥⁄ = 0. Under this circumstance, the only direction in which polymers can stretch

is the x direction. The component of �⃗⃗� in the x direction, 𝑄𝑥 ≠ 0, and the component in the y

direction, 𝑄𝑦 ≠ 0. For simplification, Q is used to replace 𝑄𝑥 in the following text. Similarly,

𝑐𝑦 = 0, 𝑐𝑥 = 𝑡𝑟𝑎𝑐𝑒(𝑐), and c is used to replace 𝑡𝑟𝑎𝑐𝑒(𝑐). T is used to replace the component of

�⃗⃗� in the x direction.

These simplifications make the problem one-dimensional, and Eq. 1-1, Eq. 1-3, and Eq.

1-5 are reduced to the following,

𝑑𝑝

𝑑𝑥=

𝛽

𝑅𝑒𝑏

𝑑2𝑢𝑥

𝑑𝑦2+

1 − 𝛽

𝑅𝑒𝑏

𝑑𝑇

𝑑𝑥 (3-6)

𝑢𝑥

𝑑𝑐

𝑑𝑥= −𝑇 = −

1

𝑊𝑖(

𝑐

1 −𝑐𝐿2

− 1)

(3-7)

33

With complete laminarization of turbulent flow explained earlier, the velocity profile is

assumed to be a second order polynomial (all the parameters used, i. e., velocity and distance, are

still dimensionless) (here the Cartesian coordinates is used since all governing equations use the

Cartesian coordinates):

𝑢𝑥 = 𝑎0𝑦2 + 𝑎1𝑦 + 𝑎2 (3-8)

In the pipeline, the following three boundary conditions can be used to obtain the

coefficients in Eq. 3-8.

𝑢𝑥(1) = 𝑢𝑥(0) = 0 (3-9)

1 = ∫ 𝑢𝑥

1

0

𝑑𝑦 (3-10)

This leads to the following velocity profile:

𝑢𝑥 = −6𝑦2 + 6𝑦 (3-11)

which is similar to the results by Japper-Jaafar et al. (2010)

According to Eq. 3-6, the pressure gradient without polymers can be represented in Eq. 3-

12 as β =1 when no drag-reducing agents are added.

(𝑑𝑝

𝑑𝑥)

𝑤𝑖𝑡ℎ𝑜𝑢𝑡=

1

𝑅𝑒𝑏

𝑑2𝑢𝑥

𝑑𝑦2 (3-12)

The definition of drag reduction (DR%) is given in Eq. 3-13.

𝐷𝑅% =(

𝑑𝑝𝑑𝑥

)𝑤𝑖𝑡ℎ𝑜𝑢𝑡

− (𝑑𝑝𝑑𝑥

)𝑤𝑖𝑡ℎ

(𝑑𝑝𝑑𝑥

)𝑤𝑖𝑡ℎ𝑜𝑢𝑡

× 100% (3-13)

(𝑑𝑝 𝑑𝑥⁄ )𝑤𝑖𝑡ℎ and (𝑑𝑝 𝑑𝑥⁄ )𝑤𝑖𝑡ℎ𝑜𝑢𝑡 represent pressure drop with and without polymers,

respectively. After Eq. 3-6 and Eq. 3-12 are introduced, Eq. 3-13 can be transformed to:

34

𝐷𝑅% = (1 − 𝛽) (1 −

𝑑𝑇𝑑𝑥

𝑑2𝑢𝑥

𝑑𝑦2

) (3-14)

The term 𝑑𝑇 𝑑𝑥⁄ , according to chain rule of derivation, can be written as:

𝑑𝑇

𝑑𝑥=

𝑑𝑇

𝑑𝑐

𝑑𝑐

𝑑𝑥=

1

𝑢𝑥𝑊𝑖2

1 − 𝑐 −𝑐𝐿2

(1 −𝑐𝐿2)

3 (3-15)

One method of establishing an average drag reduction in the pipeline is to use the

dimensionless bulk velocity (1) to replace 𝑢𝑥. Thus, the equation for average drag reduction

(DR%) can be shown as:

𝑅% = (1 − 𝛽) (1 +1

12𝑊𝑖2)

1 − 𝑐 −𝑐

𝐿2

(1 −𝑐

𝐿2)3 (3-16)

Eq. 3-16 gives the upper limit of drag reduction with polymer additives when complete

laminarization happens in a turbulent pipe flow.

It is impossible to measure the conformation tensor and maximum length of the polymer

in the flow. To validate this model and make it useful in practical engineering applications, a

special case of c = 0 is considered to yield the following,

𝐷𝑅% = (1 − 𝛽) (1 +1

12𝑊𝑖2) (3-17)

This represents an extreme condition with lowest drag reduction ability of the polymers

when the polymers are in a completely coiled state therefore the end-to-end vector is 0, which

was proved by Procaccia et al. (2008). In any other conditions when the polymers are stretched,

i.e., 𝑐 ≠ 0, the drag reduction will be higher than what is predicted by Eq. 3-17.

The remaining polymer properties in Eq. 3-17, 𝛽 and Wi, can be easily measured in a

pipe flow or in a rheometer. Then the model can be validated and further analyzed. One should

35

also note that the difference between Eq. 3-16 and Eq. 3-17. (1 − 𝑐 − 𝑐 𝐿2⁄ ) (1 − 𝑐 𝐿2⁄ )3⁄ ,

represents the effects of polymer length and conformation tensor, which are both related to

viscosity and elasticity properties of the fluid. These viscoelastic effects can also be investigated

alternatively with the remaining properties in Eq. 3-17. These aspects will be discussed in the

following section.

3.2 Model Tuning and Discussions

Since we establish an upper limit model of drag reduction by polymers, experimental data

from Japper-Jaafar et al. (2009) are used to tune the model developed in this work. Eq. 3-17 will

be tuned against these data. Based on its definition, the Weissenberg number can be expressed by

the polymer relaxation time and the shear rate:

𝑊𝑖 = 𝜆�̇� = 𝜆8𝑈𝑏

𝑑 (3-18)

where d is the diameter of the pipeline. Relaxation time can be obtained from the definition

(Malkin & Isayev, 2017):

𝜆 =𝜇

𝐺 (3-19)

G is the elasticity and is the viscosity measured from rheometers.

The viscosity at a certain shear rate can be calculated by the Carreau–Yasuda model as:

𝜇 − 𝜇0

𝜇 − 𝜇∞=

1

[1 + (𝜆𝐶𝑌�̇�)𝑎𝐶𝑌]𝑛𝐶𝑌𝑎𝐶𝑌

(3-20)

where 𝜇0 and 𝜇∞ represent the viscosity at zero shear rate and infinite shear rate; 𝜆𝐶𝑌, nCY and aCY

are three constants correlated from experimental data. Note that 𝜆𝐶𝑌 is not relaxation time but a

time parameter in the Carreau–Yasuda model. The other parameter, 𝛽, can also be calculated when

shear rate (velocity) is known.

36

In the experimental data, drag reduction values were measured with concentration of

0.075% and less, while the modulus to calculate the Weissenberg number was measured with

concentration of more than 0.075%. So, the modulus found for concentrations of less than

0.075% with high shear rate (100 s-1) will be used in all calculations as it is impossible to

estimate the modulus for concentrations of less than 0.075%. The loss and storage modulus, G’

and G”, under shear rate are 0.2 and 0.4 Pa by estimation from the original paper. Thus, the

modulus based on definition in Eq. 3-21 is 0.45 Pa, which will be used in all the following

calculation.

𝐺 = √(𝐺′)2 + (𝐺")2 (3-21)

Since the current model predicts upper limit of drag reduction (with complete

laminarization), the predicted drag reduction (with polymeric fluid properties measured from

rheometer), DRCal, should always be larger than the one reported for pipelines, DRExp. This can

be clearly seen in Figure 3-1, providing the desired tuning.

Figure 3-1 Comparison of experimental data with calculated drag reduction from the upper limit

model (data from Japper-Jaafar et al. (2009))

37

In Figure 3-1, the straight line represents DRCal = DRExp. All data points are above the

straight line, which shows that DRCal is indeed greater than DRExp over the entire range of data.

One may notice that although most DRCal values in Figure 3-1 are in the range of 0 – 1, some are

larger than 1. Real drag reduction (percentage) should always be less than 1. Those values larger

than 1 are due to the experimental data issue explained earlier. The reference by Japper-Jaafar et

al. (2009) did not provide modulus values for data conditions of concentrations less than 0.075%.

For those conditions, estimation was made using the modulus for concentration of 0.075%

(higher than their true values). The trend of data by Japper-Jaafar et al. (2009) suggests that

higher concentration leads to higher modulus. So for those conditions, the modulus used in

prediction was higher than the true values. This subsequently makes the calculated drag

reduction larger. This is evident in Eq. 3-22, which is an expansion of Eq. 3-17.

𝐷𝑅% = (1 − 𝛽) (1 +1

12 (𝜇𝐺 �̇�)

2) = (1 − 𝛽) (1 +𝐺2

12(𝜇�̇�)2) (3-22)

If complete data were available, those points for lower concentrations would have lower

modulus values, making the calculated drag reduction lower than 1. From this perspective, these

results actually verify that the developed model works well in drag reduction prediction with

fluid and polymer properties.

A new parameter for the ratio of DRCal and DRExp can be introduced,

𝛼 =𝐷𝑅𝐸𝑥𝑝

𝐷𝑅𝐶𝑎𝑙 (3-23)

Since DRCal is larger than DRExp in all experimental conditions, 𝛼 is less than 1. The

relationship between 𝛼 and DRExp is shown in Figure 3-2.

38

Figure 3-2 The linear relationship between 𝛼 and DRExp

Figure 3-2 shows an interesting linear relationship between the drag reduction ratio 𝛼 and

DRExp,

𝛼 = 0.0120𝐷𝑅𝐸𝑥𝑝 − 0.0712 (3-24)

Combining Eq. 3-23 and Eq. 3-24 then rearrangement leads to an explicit relationship

between the calculated upper limit of drag reduction and estimated drag reduction in pipe flow,

as shown in Eq. 3-25.

𝐷𝑅𝐸𝑥𝑝 =0.0712𝐷𝑅𝐶𝑎𝑙

0.012𝐷𝑅𝐶𝑎𝑙 − 1 (3-25)

It is often costly and time-consuming to build a flow loop to test the drag reduction in

pipelines, and even harder to monitor drag reduction performance in real pipelines. In this case,

Eq. 3-25 provides a convenient way to estimate drag reduction in pipeline (DRExp) from the

upper limit of drag reduction (DRCal) calculated using the analytical model and fluid properties,

39

i.e., modulus and viscosity, measured in rheometer. These relationships are depicted in Figure 3-

3.

Figure 3-3 Relationships between measured fluid properties in rheometer, calculated upper limit

of drag reduction (DRCal), and estimated drag reduction in pipe flow (DRExp)

It should be noted that this correlation may be dependent on the types of chemical

additives, but a similar method can be used for other DRAs, which may result in a different

coefficient for the linear relationship in Eq. 3-24. This study will not attempt to develop a

universal correlation. However, if more studies combining rheometer and flow loop

measurements can be completed, the coefficients in Eq. 3-24 can be refined, making this

correlation applicable to a wider range of conditions of drag reduction in pipelines.

In this method, a relaxation time of polymer is needed to predict the drag reduction.

However, when the polymer concentration is extremely low, it may be impossible to measure the

relaxation time (Lim et al., 2003). In this case, one method to predict the Deborah number, a

dimensionless number regarding relaxation time, can be used (Hong et al., 2015).

Further examination of the model in Eq. 3-17 leads to useful insights to understand the

drag reduction mechanism, particularly the effects of elasticity and viscosity. Expanding Eq. 3-

17 also gives the following:

𝐷𝑅% =1

1 +𝜇𝑃,0

𝜇𝑆

(1 +𝐺2𝑑2

768𝑈𝑏2𝜇2

) (3-26)

Rheometer measurement (𝜇 and

G)

Eq. 3-22

Model (DRCal)Eq. 3-

25Pipe flow (DRExp)

40

Many factors are involved in Eq. 3-26, i.e., viscosity, modulus, velocity, and

concentration and they are all interrelated. It is unreasonable to evaluate the elasticity and

viscosity effects separately as both can contribute to drag reduction. Here a new explanation of

viscoelasticity effects in drag reduction is shown.

Assume that a polymer solution with a constant concentration can reduce the friction in

pipelines and the concentration varies in different conditions. At an initial state, the velocity of

flow is low then gradually increases. The total value 𝐺2 𝑈𝑏2𝜇2⁄ starts to increase, since the

growth momentum of G2 is greater than that of 𝑈𝑏2𝜇2. Thus, drag reduction increases until the

maximum drag reduction is achieved. In this period, elasticity plays a positive role since it

offsets the growth momentum of 𝑈𝑏2𝜇2, which results in a drag reduction increase. But this trend

will stop when G2 starts to grow slower than 𝑈𝑏2𝜇2. Now, 𝐺2 𝑈𝑏

2𝜇2⁄ starts to decrease, which

leads to a decrease in drag reduction. In this period, modulus plays a negative role in drag

reduction. Even though elasticity still increases, it cannot offset the growth of viscosity. Overall,

this phenomenon can be summarized as follows. At a critical concentration, drag reduction will

reach its maximum value when velocity increases to a certain point, after which, drag reduction

starts to decrease. This theory can be supported by many previous results (Gasljevic et al., 1999;

Zhang et al., 2005; Kamel & Shah, 2009; Dosunmu & Shah, 2014) shown in Figure 3-4.

41

Figure 3-4 Drag reduction performance of two DRAs ability at the same velocity originally from

Abubakar et al. (2014) and at the same concentration from Kamel & Shah (2009)

It is also important to consider the other situation: drag reduction with a constant velocity

and varying concentrations. With increasing concentration, 𝜇𝑃,0 increases and the term

1 − 𝜇𝑆 (𝜇𝑆 + 𝜇𝑃,0)⁄ increases. And increasing concentration will also increase the modulus,

which can increase the drag reduction. When the concentration reaches a certain point, the drag

reduction also reaches its maximum, after which, drag reduction starts to decrease even though

elasticity still increases with increasing velocity. The increasing momentum of modulus cannot

offset the momentum of the viscosity, so drag reduction starts to decrease. This phenomenon has

also been observed by previous studies (Abubakar et al., 2014; Guersoni et al., 2015) as shown in

Figure 3-4.

42

3.3 Summary

In Chapter 3, a new model for drag reduction by polymer is proposed and tuned with

previous experimental data. This model assumes complete laminarization in the flow and

predicts the upper limitation of drag reduction in pipe flows. Comparison of predicted and

measured data also leads to a correlation between drag reduction in pipe flow and predicted

upper limit of drag reduction using fluid properties measured in a rheometer. The correlation

provides a convenient and useful way for estimation of pipeline drag reduction with low cost.

This model is also used to explain the mechanism of drag reduction by DRAs. Both viscosity and

elasticity of the polymeric fluid affect the drag reduction. With a constant concentration, drag

reduction increases with increasing velocity as the growth of modulus is limited. After maximum

drag reduction occurs, drag reduction starts to decrease as the growth of velocity and viscosity is

larger than that of the modulus. With a constant velocity, increasing concentration increases the

drag reduction due to positive effects from the modulus. Once a maximum drag reduction occurs,

drag reduction decreases because the growth of modulus is greater than that of the viscosity.

43

Chapter 4 Experimental Correlation for Pipe Flow Drag Reduction

Using Relaxation Time

A semi-analytical model was presented in the earlier chapter for the drag reduction

prediction by polymers. However, it is difficult to measure the relaxation time of the polymer in

the dilute solution. In this chapter, the concept of relaxation time of the polymers is further

examined and an experimental study on drag reduction in a pipe flow is conducted to develop a

correlation for drag reduction by polymers. The main content of this chapter has been published

(Zhang, X., Duan, X., Muzychka, Y., & Wang, Z. (2020). “Experimental correlation for pipe

flow drag reduction using relaxation time of linear flexible polymers in a dilute solution”. The

Canadian Journal of Chemical Engineering, 98(3), 792-803.). The author of this thesis is the first

author of this paper. The first author conducted the experiments, analyzed the data, developed

the correlation and prepared the manuscript. Prof. Duan and Prof. Muzychka as the second and

third authors provided their suggestions on the correlation development and revisions of this

paper. Prof. Wang as the fourth author helped me in the construction of the flow loop.

4.1 Theories and Correlation Formulation

4.1.1 Clarification of the Relaxation Time, Deborah Number, and Weissenberg Number

As discussed earlier, the relaxation time of a DRA is a key property in determining its

drag reduction efficiency. It refers to the transition time of a DRA from the stretched state to the

coiled state. Figure 4-1 illustrates the coiled state and stretched state of a long chain linear

flexible polymer in a dilute solution. In the coiled state, its radius of gyration, RG, refers to the

radius of an imaginary sphere enclosing the coiled polymer. This definition of relaxation was not

followed by previous works (Ghajar & Azar, 1988; Kwack & Hartnett, 1983).

44

Figure 4-1 Polymer behaviors in a dilute solution

The relaxation time could be measured by rheometers, defined as the ratio of viscosity

over the modulus. In previous studies, the relaxation time of surfactants was measured by this

method. However, this method cannot be used in dilute polymer solutions because the rheometer

cannot measure the modulus of the dilute polymer solution (Campo-Deano & Clasen, 2010). One

previous study showed that the relaxation time could be measured by a filament method using a

commercial capillary breakup extensional rheometer (CaBER) (Owolabi et al., 2017). However,

the method is still limited to relatively high concentrations, at least semi-dilute solutions (several

hundred ppm), and cannot be used for dilute solutions (<100 ppm).

Therefore, another method to investigate the relaxation time of polymers is needed in

dilute solutions. In this study, the theory of Zimm, a theory to determine the relaxation process of

linear flexible polymers, is used for this estimation and shown in Eq. 4-1 and Eq. 4-2 (Zimm,

1956). This theory appeared in many classic reviews on drag reduction (Sreenivasan, K. R., &

White, 2008; Sreenivasan & White, 2000):

𝑅𝐺 = 𝑎𝑁0.6 (4-1)

𝜆 =𝜇𝑆𝑅𝐺

3

𝑘𝐵𝑇 (4-2)

where RG is the radius of gyration of the polymers in a dilute solution (m); kB is the Boltzmann

constant, 1.38 × 10-23 J · K -1; T is the temperature (K); a is the length of monomer of polymer

45

(m); and N is the degree of polymerization. The radius of gyration is defined as the average

distance from the centre of the polymer to the edge of an imaginary sphere covering the outer

edge of the coiled polymer, as shown on the left side of Figure 4-1.

In this chapter, the concentration effect on the relaxation time of the dilute polymer

solution is not considered as it was in a previous work (Muthukumar, 1984). This concentration

dependency is due to the viscosity variation of the polymer solution. It was reported by

Ebagninin et al. (2009) that for a 4 × 106 g/mol PEO solution at a concentration of 2500 ppm, the

viscosity remains almost the same as water. In the current work, the concentration of polymer is

so low (from 5 ppm to 20 ppm) that the viscosity of the dilute polymer solution is assumed the

same as the solvent, water.

The original definition of Deborah number by Reiner (1964) was based on the ratio of relaxation

time over observation time, shown in Equation (3) (Dealy, 2010; Poole, 2012):

𝐷𝑒 =

𝜆

𝑡𝑃=

𝜆

𝑙𝑈𝑏

(4-3)

where λ is the relaxation time (s); and tP is the observation time (s), calculated from the length of

the pipe, l (m), and the average velocity, Ub (m/s). The current study considers this as the only

correct definition of this dimensionless number. Compared with the other complex definitions of

Deborah number in the literature, the definition in Eq. 4-3 has a clear physical meaning and is

easy to use.

An alternative dimensionless number involving the relaxation time is the Weissenberg

number, defined in Eq. 4-4 (Dealy, 2010; Poole, 2012):

𝑊𝑖 = 𝜆�̇� = 𝜆8𝑈𝑏

𝑑 (4-4)

46

where �̇� is the shear rate at the wall of the pipe in turbulent flow (s-1) as suggested by Metzner

and Park (1964). The shear rate in the near wall region is used because the vortex structure in this

region is modified by the polymers added in the turbulent flow (White et al., 2004). In Eq. 4-4,

Ub is the average velocity (m/s) and d is the diameter of the pipeline (m). As discussed earlier,

the viscosity of a dilute polymer solution remains the same as the solvent (water). This solution

shows Newtonian fluid characteristics, which explains the shear rate expression in Eq. 4-4. The

Weissenberg number describes the ratio of elastic force over viscous force in the flow.

These definitions of the Deborah number and the Weissenberg number, and the

difference between the two, have been discussed by Dealy (2010) and Poole (2012). Many

previous studies did not follow these definitions, and the Deborah and Weissenberg numbers are

often misused. In a pipe flow, the Deborah number (De) and Weissenberg number (Wi) are

related to the length and the diameter of the pipe, respectively. In this study, the Weissenberg

number is used since shear rate, related to the pipe diameter, is important in the drag reduction.

4.1.2 Correlation Formulation

When a polymer is used as a DRA in a turbulent pipe flow, the drag reduction efficiency,

DR%, can be defined as the relative decrease of pressure drop:

𝐷𝑅% =Δ𝑃𝑆 − Δ𝑃𝑃

Δ𝑃𝑆× 100% (4-5)

where ΔPS (Pa) is the pressure drop without the polymer DRA; and ΔPP (Pa) is the pressure drop

with the DRA. In this definition, DR% is a dimensionless number. As discussed earlier, it is

determined by several parameters. With a constant temperature, the drag reduction efficiency is a

function of the pipe diameter (d), bulk velocity (Ub), viscosity of the solvent (µS), and the

characteristic length of the turbulent flow with polymers. This length is a function of the polymer

type, molecular weight, and concentration (Gasljevic et al., 1999). In this study, only water is

47

used as the solvent and its density is ~1000 kg/m3; therefore, the density is not added as a

variable. In this experiment, only one type of polymer (PEO) is used, and therefore the degree of

polymerization (N) is used to replace the average molecular weight, M. The polymer

concentration (CP) is essentially a dimensionless number (weight-based, ppm). With these

considerations, the drag reduction can be expressed in Eq. 4-6:

𝐷𝑅% = 𝑓(𝑑, 𝑈𝑏 , 𝜇𝑆, 𝐶𝑃, 𝑁) (4-6)

Eq. 4-6 has six factors and three dimensions (length, time, and mass). To make Eq. 4-6

dimensionless, three dimensionless groups using these variables should be prepared. Two

dimensionless numbers already exist, DR% and CP. The remaining question is how to combine

the other four variables, d, Ub, N, and 𝜇𝑆 to make one dimensionless number, the details of which

can be found in Appendix I. Examining the definition of the Weissenberg number in Eq. 4-4, and

the relaxation time estimation from Eq. 4-1 and Eq. 4-2, one can find that the diameter d,

velocity Ub, dynamic viscosity 𝜇𝑆, and polymerization degree N can all be incorporated in the

Weissenberg number. Thus, Eq. 4-6 becomes the following:

𝐷𝑅% = 𝑓(𝑊𝑖, 𝐶𝑃) (4-7)

Many formats exist for the drag reduction correlation (Koskinen et al., 2004; Shah &

Vyas, 2011; White, 1970), with different degrees of acceptance in the literature. In this work, the

findings in previous studies are followed to develop the format of Eq. 4-8 for the prediction of

drag reduction from a viscoelastic perspective. In Eq. 4-8, the viscoelastic prosperity is separated

into two parts, viscous part and elastic part. The elastic part is the square of the Weissenberg

number, Wi2, suggested in our previous work (see appendix II for further explanations about why

Wi2 is used instead of Wi) (Zhang et al., 2018); the viscous part is the polymer concentration, CP,

based on work by Kim et al. (1997) and Yang et al. (1994). With substantial experimental data,

48

these researchers showed that in a dilute polymer solution the drag reduction was approximately

proportional to the polymer concentration, i.e., 𝜕𝐷𝑅 𝜕𝐶𝑃 ≈ 𝑐𝑜𝑛𝑠𝑡⁄ . The other experimental

conditions incorporate molecular weight, velocity, and geometry in the drag reduction flow,

which are combined as the Weissenberg number mentioned above. Without polymer, CP = Wi =

0, there will be no drag reduction, i.e., DR% = 0. The two constants, A and B, in Eq. 4-8 are to be

determined from experimental data:

𝐷𝑅% = 𝐴𝐶𝑃 + 𝐵𝑊𝑖2 (4-8)

4.1.3 Concentration Range for Correlation

This correlation, Eq. 4-8, works for drag reduction in dilute polymer solutions as

mentioned. Under this condition, there is no interaction between two polymer chains, as

illustrated in Figure 4-2a, and the only interaction is the one between the solvent and the

polymer. Here it is helpful to understand the concept of overlap concentration for polymer

solutions. Figure 4-2 illustrates relationships between polymer chains (in coiled state and shown

as imaginary spheres) in solutions of three different concentrations. In the dilute solution, the

concentration CP is low and there is no overlap or interaction between the polymers, shown in

Figure 4-2a. At a critical concentration, C*, the imaginary spheres of polymers are tangent to

each other, as shown in Figure 4-2b; there will be interactions between them once the

concentration increases slightly. This critical concentration is called the overlap concentration

(Cotton et al., 1976; Graessley, 1980; Ying & Chu, 1987). If the concentration continues to

increase, there will be significant interactions between the polymer chains and the solution is

called a semi-dilute polymer solution, shown in Figure 4-2c.

49

Figure 4-2 (a) Dilute polymer solution when c is less than C*, (b) Critical state when c is equal to

C*, (c) Semi-dilute solution when c is greater than C*

This overlap concentration concept has been discussed in many publications (Graessley,

1980; Ying & Chu, 1987), and many equations have been proposed to calculate the overlap

concentration. In this study, one classic equation by Broseta et al. (1986) and Cotton et al. (1980)

is used, as shown in Eq. 4-9:

𝐶∗ =𝑀

43 𝜋𝑁𝐴𝑅𝐺

3 (4-9)

where M is the average molecular weight, g·mol-1; and NA is the Avogadro constant, 6.02 × 1023

mol-1. Note that there are other methods to calculate the overlap concentration, such as those

based on the sudden change of viscosity versus the polymer concentration (Ebagninin et al.,

2009). The definition in Eq. 4-9 is better since it has clear physical meaning and is widely

accepted.

In this study, PEO (from Sigma-Aldrich Canada) as the drag-reducing polymer is used,

with three viscosity-average molecular weights, 106, 2 × 106, and 4 × 106 g/mol. The viscosity-

average molecular weight to calculate the degree of polymerization (N) is used because it is the

only available molecular weight from the supplier of these polymers. The monomer length to

calculate the radius of gyration of relaxation time of PEO is 0.278 nm (Oesterhelt et al., 1999).

(a) (b) (c)

50

The overlap concentration is shown in Table 1. The profile of the molecular weights is not

considered since this study intends to use the average molecular weight to predict the drag

reduction via relaxation time.

Table 4-1 Overlap concentration (C*) at different molecular weights

M (106 g/mol) N RG (× 107 m) C* (g/m3 ≈ ppm)

1 22727 1.14 266

2 45455 1.73 153

4 90909 2.63 88

The Flory interaction (or solubility) parameter is also an important parameter of polymers

in drag reduction, as suggested by Choi et al. (1999) and Lim et al. (2007). However, the current

correlation of Eq. 4-6 does not include this parameter. It is not a variable in our study since only

one type of drag reducing polymer (PEO) is used and it has a fixed solubility parameter. This

variable will be considered in future research that involves other polymer types. Since the PEO

concentration is low, it is assumed that the density of the dilute solution remains the same as the

water density, ~103 kg/m3 in experimental conditions. Thus, g/m3 is treated as g/106 g, the latter

being equal to ppm (here ppm is dimensionless). Thus, the concentration in this study should be

less than 88 ppm. Even lower concentrations are used in the experiments to ensure dilute

solutions.

4.2 Experimental Setup and Procedure

The schematic of the flow loop for the drag reduction investigation is illustrated in Figure

4-3. Water is transported to the pipeline via a self-prime pump (6050 Series Bronze AC Motor

51

Pump Unit, Xylem USA) from a 0.22 m3 tank. To regulate the flow rate in the experiment, two

globe valves are installed in the main pipeline with bypass. A high-accuracy flowmeter

(FTB691A-NPT from Omega, USA) measures the flow rate in the experiment. The uncertainty

of flow rate measurement is as follows: 3% of reading for flow rate from 3.8 L/min-38 L/min

(litre per minute), and 5% when the flow rate is less than 3.8 L/min. The nominal pipe diameter

from the outlet of the pump to the polymer injection point is 0.0254 m.

Figure 4-3 Schematic diagram (a) and photo (b) of the flow loop

The concentrated PEO solution in the polymer solution tank is prepared in a mild

condition. Tap water and PEO are mixed with a magnetic stirring device in a low rotational

(a)

(b)

52

speed for 24 hours until no cluster or aggregate can be observed. This concentrated PEO solution

is settled for another 24 hours to be homogenized and it will be used up in one day. The

concentrated polymer solution is injected into the pipe flow by a diaphragm metering pump

(PHP-804M, Omega, USA). Previous studies showed that this heterogeneous injection method

with a diaphragm pump prevented mechanical degradation of the DRAs and led to optimal drag

reduction performance (Hoyer & Gyr, 1998; Hoyt & Sellin, 1988; Vleggaar & Tels, 1973; Wells

Jr & Spangler, 1967). Also, the master solutions are injected at the wall (rather than centre) of

the pipe as suggested by previous studies for better drag reduction efficiency (Hoyt & Sellin,

1988; Kim & Sirviente, 2007). The concentration of all PEO concentrated solutions (with PEOs

of three molecular weights) is 750 ppm, and the concentration in the pipeline is 5, 10, 15, and 20

ppm with four different flow rates.

After the injection point, a stainless-steel pipe with an inner diameter of 1.27 cm is used

as the test section. To eliminate the entrance effect in drag-reducing pipe flow, the entrance

length before the test section is 1.3 m (~102d), as suggested by previous studies (Omrani et al.,

2012; Seyer & Catania 1972; Tuan & Mizunuma, 2013). The pressure drop in the test section

(2.02 m long, ~159d) is measured by a differential pressure transducer (DPGM409-350HDWU,

Omega, USA). The measurement uncertainty for the pressure drop is 28 Pa. The polymer

solutions are collected at the outlet of the test section for treatment and disposal to avoid

environmental problems. Temperature is measured in all experiments for the calculation of the

Weissenberg number.

The Fanning friction factor can then be calculated from the experimental data using Eq.

4-10:

53

𝑓 =

𝑑Δ𝑃

2𝑙𝜌𝑢2 (4-10)

where d is the diameter of the pipe; l is the length; ρ is density of the dilute polymer flow (as

explained earlier); ΔP is the measured pressure drop; and u is the mean velocity calculated from

the measured flow rate from Eq. 4-11:

𝑢 =

4𝑄

𝜋𝑑2 (4-11)

Using the methods of Kline and McClintock (1953) the uncertainty of the friction factor

can be estimated with Eq. 4-12:

𝛿𝑓 = √(𝜕𝑓

𝜕(Δ𝑃)𝛿(Δ𝑃))

2

+ (𝜕𝑓

𝜕𝑢𝛿𝑢)

2

+ (𝜕𝑓

𝜕𝑑𝛿𝑑)

2

+ (𝜕𝑓

𝜕𝑙𝛿𝑙)

2

(4-12)

where 𝛿(ΔP) is the uncertainty of the differential pressure sensor; 𝛿d and 𝛿l are 0.1 mm and 1

mm, respectively; and 𝛿u is the uncertainty the measured mean flow velocity, which can also be

estimated using the same Kline and McClintock method:

𝛿𝑢 = √(𝜕𝑢

𝜕𝑄𝛿𝑄)

2

+ (𝜕𝑢

𝜕𝑑𝛿𝑑)

2

(4-13)

where 𝛿Q is the uncertainty of flow rate measurement as discussed earlier. Similarly, the

uncertainty of drag reduction efficiency (DR%) can be estimated with Eq. 4-14:

𝛿(𝐷𝑅%) = √(𝜕(𝐷𝑅%)

𝜕(Δ𝑃𝑆)𝛿(Δ𝑃𝑆))

2

+ (𝜕(𝐷𝑅%)

𝜕(Δ𝑃𝑃)𝛿(Δ𝑃𝑃))

2

(4-14)

A series of experiments were conducted with water but without polymer additives in the

test section to verify the experimental setup. For verification purposes, the measured Fanning

friction factors are compared with the classic Colebrook–White correlation in Eq. 4-15:

54

1

√𝑓= −2 log (

2.51

𝑅𝑒√𝑓+

휀

3.7𝑑) (4-15)

where Ɛ is the absolute roughness of the pipe, equal to 0.015 mm in this study. The results in

Figure 4-4 show a good agreement between the measurements and the results from the classic

correlation: all predicted value are covered by the error bars from the uncertainty analysis. This

demonstrates the reliability of the experimental setup and data analysis method, and guarantees

the data repeatability due to the small error. Note that error bars are shown in Figure 4-4 yet not

shown in the figures of the next sections, for the purpose of clean presentations.

Figure 4-4 Benchmark test for the flow loop

4.3 Experimental Results

4.3.1 Drag Reduction Experimental Data and Analysis

Figure 4-5 shows the measured Fanning friction factor at different Reynolds numbers of

the drag-reducing flow. As clarified earlier, the solvent viscosity is used to calculate the

55

Reynolds number since in the dilute polymer solution (several ppm levels), the polymers will not

significantly change the viscosity (Burshtein et al., 2017). As expected, the friction factors with

polymers are less than the friction factors predicted by the Blasius equation (4000 ≤ Re ≤ 4×104

in pipe flows) with no drag reduction, i.e., Eq. 4-16 and the dash-dotted lines in Figure 4-5. Also,

the friction factors of flow with polymers are higher than the friction factors predicted by Virk’s

asymptote (Virk, 1975; Virk et al., 1970, 4000 ≤ Re ≤ 4×104 in pipe flows), i.e., Eq. 4-17 and the

dotted lines in Figure 4-5, which represents the minimum friction factor by polymers. The

laminar flow and laminar flow extension lines from the Hagen–Poiseuille equation, i.e., Eq. 4-18

(Re ≤ 2300 in pipe flows) and solid and dash lines in Figure 4-5, are used as a reference to

indicate that whatever polymer is added to reduce the friction, it is always greater than the ones

in a laminar flow:

𝑓 = 0.079𝑅𝑒−0.25 (4-16)

𝑓 = 0.59𝑅𝑒−0.58 (4-17)

𝑓 =

16

𝑅𝑒 (4-18)

These qualitative analyses further verify that experimental data are correct and also show

that the drag reduction is influenced by polymer concentration.

56

(a)

)

(c)

(b)

)

57

Figure 4-5 The relationship between friction factor and Reynolds number at different


Figure 4-6 shows the relationship between the drag reduction and polymer concentration.

With the same polymer, a higher concentration can increase the drag reduction. In a polymer

solution with a higher concentration, more polymers are available to dampen the turbulent

structures. The energy dissipated by these turbulent structures can be reused for flow in the

stream-wise direction. In this case, the turbulent flow needs less extra energy (pressure drop) to

sustain. This is how the drag reduction is defined (Benzi & Ching, 2018).

(a)

58

Figure 4-6 The relationship between drag reduction and Reynolds number at different


Figure 4-7 shows the relationship between the Weissenberg number and drag reduction.

The sloid straight lines in Figure 4-7 represent the linear relationship between the drag reduction,

DR%, and the square of the Weissenberg number, Wi2, as suggested in a previous work (Zhang

(b)

(c)

59

et al., 2018). Under a given molecular weight, the drag reduction increases with an increasing

Weissenberg number. As discussed earlier, a higher Weissenberg number means a higher

elasticity in the solution, which is better in dampening the vortex structure in the flow so the

energy can be used for flow in the streamwise direction (De Gennes, 1986; Tabor & De Gennes,

1986).

(a)

(b)

60

Figure 4-7 Drag reduction at different Weissenberg numbers: (a) when M = 106 g/mol; (b) when

M = 2 × 106 g/mol; (c) when M = 4 × 106 g/mol

4.3.2 Correlation and Validation

All experimental data are analyzed in Excel and a regression is conducted following the

format in Eq. 4-8, leading to the following final correlation:

𝐷𝑅% = 0.935𝐶𝑃 + 0.858𝑊𝑖2 (4-19)

The coefficient of determination of Eq. 4-19, R2, is 0.96, indicating a good correlation for

the experimental data. As shown in Figure 4-8, most data are adequately covered within a +/-

30% relative error range.

To further validate the correlation and demonstrate its wider range of applications, other

drag reduction data by PEO from previous works are added in the analysis and shown in Figure

4-8 (Goren & Norbury, 1967; Inge et al., 1979; Interthal & Wilski, 1985; Kim et al., 2009;

Paterson & Abernathy, 1970). These additional data are obtained in other conditions, i.e.,

(c)

61

different molecular weight, concentration, flow velocity, and diameter, as summarized in Table

4-2. All experiments were conducted at room temperature, ~17 °C.

Table 4-2 Summary of previous experimental data and conditions for correlation validation

Reference M (106 g/mol) CP (ppm) u (m/s) d (cm)

Goren & Norbury (1967) 4 5-20 0.73-1.11 5.1

Inge et al. (1979) 4 0.4-1.6 1.44-2.17 1

Interthal & Wilski (1985) 4 50 0.74-2.04 15.2-20.3

Kim et al. (2009) 0.2-4 1-20 2-3.3 1.71

Paterson & Abernathy (1970) 0.5 5-10 2.67-4.15 0.63

Figure 4-8 Comparison of drag reduction data from experiments and predictions by the

developed correlation in Eq. 4-19

62

As shown in Figure 4-8, these extra data are also in the +/- 30% range of predictions from

Eq. 4-19. These results show that the new correlation has a wide range of applications and is not

limited to conditions in this study. In these additional validations, the most convincing one is

from the work of Interthal and Wilski (1985). In this work, the test was conducted in a pipe

system with 15.2 and 20.3 cm diameters, the latter being the largest diameter in the drag

reduction tests by PEO until the current study, which is very close to pipes in real liquid

transportation systems.

4.4 Discussions

The theory and correlation developed in this work can explain an interesting phenomenon

previously discovered by Peyser & Little (1971). They mentioned that an increasing viscosity of

the solvent led to an increase in the drag reduction. If the definitions of Weissenberg number in

Eq. 4-4 and relaxation time in Eq. 4-2 are combined, one can see that a higher viscosity leads to

higher relaxation time and Weissenberg number. From the new correlation in Eq. 4-19, a higher

Weissenberg number leads to higher drag reduction.

The correlation developed in this study provides a promising tool for predicting drag

reduction with lower concentration polymer additives. One advantage of this new correlation

over the previous ones is that it includes the key property of relaxation time in a non-dimensional

form; therefore, the polymer type and molecular weight are considered. Many earlier drag

reduction studies used another method, the power law of drag-reducing flows, to correlate

experimental data (Bogue & Metzner, 1963; Dodge & Metzner, 1959; Malin, 1997). That

method does not require an understanding of the relaxation process. Instead, it only requires

information about the flow consistency index and flow behaviour index. There is an obvious

flaw in this method. If the polymer concentration is so low that these two factors are the same as

63

those for a Newtonian fluid, this method will no longer work. The new method of using the

relaxation time in the current study overcomes this problem. Even in a dilute polymer solution,

the relaxation time can still be estimated by the Zimm’s theory.

An additional advantage is that there is no need to build a large-scale flow device or use a

high flow velocity by this new correlation. In most previous correlations for the drag reduction

estimation, the friction factor was a function of Reynolds number (Hoyt & Sellin, 1993; Liang et

al., 2017). In the Reynolds number definition, the diameter and velocity are the numerators. Most

industrial drag reduction applications involve large pipes and higher flow velocity. To achieve

these large Reynolds numbers with a small pipe in the lab, the velocity must be increased

dramatically. The high velocity and large pressure drop requires expensive instruments and can

cause many safety problems. The new correlation in this study does not use the Reynolds number

and uses the Weissenberg number instead to predict the drag reduction. In the Weissenberg

number definition, the pipe diameter is the denominator and the velocity is the numerator. To

predict the flow drag reduction in a pipe with a large diameter, one can calculate the ratio of

velocity and diameter, and use this ratio in a flow loop with a small diameter to predict the

reduced friction. Thus, this method can predict the drag reduction based on the industrial needs

and avoid the lab safety and cost problems mentioned above. As shown in the correlation

validation, previous data are used to validate the correlation. The diameters in these tests are

different from the one used in our experiments, which range from 0.63 cm-20.3 cm. The

validation result is acceptable, within a +/- 30% relative error, confirming that one can use a

small flow loop for the prediction of the reduced friction at a larger scale. To further validate our

model, future drag reduction tests can be conducted in smaller diameter pipes (< 1 mm) as

suggested by Ushida et al (2012, 2016 & 2018).

64

When this new correlation is used for long-distance fluid transportation systems, the

degradation issue needs to be considered. In these long-distance systems, the polymers tend to

degrade over time under a high shear rate. Nesyn et al. (2018) showed that in a long-distance oil

transportation system, when a proper drag-reducing polymer was selected, the ratio of drag

reduction at the 500th km and 50th km is ~0.56. This empirical ratio may be useful in the design

of drag reduction for long pipeline transportation. The study of the degradation of DRAs is

another important research topic, and it will be discussed in the next chapter.

4.5 Summary

In Chapter 4, a semi-empirical correlation based on two dimensionless numbers, the

Weissenberg number and polymer concentration, is developed for the estimation of the drag

reduction in turbulent pipe flow in dilute polymer solutions. The Weissenberg number in the

correlation is based on Zimm’s theory regarding the polymer relaxation process, and the polymer

concentration is lower than the overlap concentration. Experimental data from the flow loop

shows that the drag reduction efficiency increases with the Weissenberg number and polymer

concentration. These data are used to develop the correlation, and the relative error between

experimental data and predicted values is within +/- 30%. Previous experimental data in

laboratory and industrial systems are used to further validate the correlation with similar good

agreement. The application and advantages of this new correlation is discussed. One of the key

benefits is that one can use a lab-scale flow loop in relatively low velocities and small diameters

to achieve the prediction of the drag reduction in a large industrial scale system.

65

Chapter 5 Mechanism and Correlation for Degradation of Drag

Reduction by Polymers in Rotational Flows

In this chapter, the degradation of drag reduction is investigated and a mechanism of the

degradation is proposed. The main content of this chapter has been published (Zhang, X., Duan,

X., & Muzychka, Y. (2018). “New mechanism and correlation for degradation of drag-reducing

agents in turbulent flow with measured data from a double-gap rheometer”. Colloid and Polymer

Science, 296(4), 829-834). The author of this thesis is the first author of this paper. The first

author conducted the experiments, analyzed the data, and prepared the manuscript. Prof. Duan

and Prof. Muzychka as the second and third authors provided their suggestions on correlation

development and revision of this paper.

5.1 Experiment

The DRA used in this study is a water-soluble polymer - PEO (from Sigma-Aldrich

Canada) with three molecular weights, 106, 2×106, 4×106 g/mol. First the polymers are mixed

with deionized water under a mild rotation speed, 60 rpm, for 5 hours until all the polymers are

dissolved. Then, the solution is left to stand still for 24 hours to ensure homogeneous

concentration. No clustering or crystallization was observed in the polymer solution before each

experiment. The concentration of polymer used in the experiment is 20, 35 and 50 ppm.

The modified RDA used in this study is based on a MCR301 rheometer (Physica, Anton Paar

Ltd, United Kingdom) with a high torque resolution, 10-6 N·m. The temperatures in experiments

were maintained at constant levels of 25, 45 and 65 oC with a high-resolution temperature-

control system. The geometry used in our experiments is a double gap (DG) shown in Figure 5-1.

There are two advantages with this geometry. First, it can reduce the evaporation of the liquids

under relatively high temperatures. This helps to avoid measurement error due to change of

66

concentration of the polymer solution, since a fixed concentration is required in each experiment.

Therefore, this DG design is more suitable than the other geometries, such as parallel plate (PP)

and cone plate (CP). Second, this geometry can also prevent samples splashing out. In the test of

drag reduction and degradation, a high shear rate (rotation speed) is needed (Andrade et al.,

2016). If the CP or PP were used in the test, the sample could splash out resulting in a

measurement error. It is believed that a DG device is the best choice for the drag reduction and

degradation test (Pereira et al., 2013).

Figure 5-1 (a) Geometry of a double gap (DG) rheometer, R1 = 11.909 mm, R2 = 12.328 mm, R3

= 13.332 mm, R4 = 13.797 mm; (b) photo of the rheometer (The copyright of this photo belongs

to the Anton-Paar website)

(a) (b)

67

Drag reduction is measured by the torque with (TP) and without DRAs (TS) under the

same rotation speed, temperature, and concentration of the polymer, as the follow:

𝐷𝑅 =

𝑇𝑆 − 𝑇𝑃

𝑇𝑆 (5-1)

The uncertainty analysis is carried out based on the Kline & Mcclintocks’s theory (1953),

shown in Eq. 5-2.

𝛿(𝐷𝑅) = √[𝜕(𝐷𝑅)

𝜕𝑇𝑆𝛿𝑇𝑆]

2

+ [𝜕(𝐷𝑅)

𝜕𝑇𝑃𝛿𝑇𝑃]

2

(5-2)

In Eq. 5-2, 𝛿(DR), 𝛿TS and 𝛿TP are the uncertainty of drag reduction, torque without

polymer solutions and with polymer solutions. The latter two terms, 𝛿Ts and 𝛿Tp are equal to

0.001 mN·m. These two terms can be replaced by 𝛿T. Eq. 5-2 can be simplified to Eq. 5-3

𝛿(𝐷𝑅) =𝛿𝑇

𝑇𝑆

√1 + (𝑇𝑃

𝑇𝑆)

2

(5-3)

Because TP is always smaller than TS in drag reduction experiments, the ratio of TP/TS is

less than 1. In these experiments, the minimum value of TS is approximately 1 mN·m and the

uncertainty, 𝛿T, in this rheometer is 0.001 mN·m. The maximum uncertainty of drag reduction

𝛿(DR) is 0.0014, which is far less than the drag reduction range in the experiment. So, the

uncertainty in the following experiments can be neglected and error bars are not shown in the

figures of results.

5.2 A New Theory of Degradation Mechanism by Polymers

As discussed in the introduction, the current theory cannot explain the mechanism of drag

reduction degradation since the physical meanings of many factors are still not clear. Here a new

theory is proposed by analyzing the degradation of DRA from a chemistry dynamics perspective.

Bizotto & Sabadini (2008) used the rate of DRA concentration change, with a unit of

68

mol·m-3·s-1, to represent the degradation of DRA in a rotating disk apparatus experiment. This is

a unit of chemical reaction. Inspired by this unit, it is proposed that DRA degradation is treated

as a chemical reaction. The original polymer concentration decreases due to the turbulent flow. It

means that original polymers with a high molecular weight are cut into smaller entities and new

polymers with low molecular weights form. (𝑟𝑝 = 𝑘 𝑑𝐶𝑝 𝑑𝑡⁄ , k is the chemical reaction constant

and 𝑑𝐶𝑝 𝑑𝑡⁄ is the change rate of polymer concentration). However, DRA degradation is not

only related to its concentration, but also the molecular weight. Several previous non-drag-

reduction studies (Madras & Chattopadhyay, 2001; Sung et al., 2004) used the change of

polymer’s molecular weight to represent the degradation rate of polymers. This view, which uses

molecular weight to indicate the degradation and drag reduction to study polymer degradation in

the drag reduction, is used. Following this practice, the present work considers DRA degradation

as a chemical reaction with a reaction rate represented by the change of molecular weight, shown

in Eq. 5-4,

𝑟𝑃 = 𝑘[𝑀]𝛼 (5-4)

where rP is the reaction rate, i.e., DRA degradation rate, with a unit of g·mol-1·s-1 ; [M] is the

molecular weight of DRA, with a unit of g mol-1; and α is the order of degradation rate; k is the

reaction constant. In chemistry, the reaction rate by definition is

𝑟𝑃 = −

𝑑[𝑀]

𝑑𝑡 (5-5)

Combining Eq. 5-4 and Eq. 5-5 leads to

−

𝑑[𝑀]

𝑑𝑡= 𝑘[𝑀]𝛼 (5-6)

Rearranging and integration leads to Eq. 5-6:

69

− ∫

𝑑[𝑀]

𝑑𝑡

𝑀(𝑡)

𝑀(0)

= ∫ 𝑘𝑑𝑡𝑡

0

(5-7)

M(0) and M(t) are the molecular weight at initial state (t = 0) and any time, t, during the

degradation.

A widely-accepted assumption proposed by Brostow and his colleagues (1983 & 1990) is

shown in Eq. 5-8, i.e., the ability of drag reduction is proportional to the molecular weight of

DRA. Here DR(0) and DR(t) is the drag reduction at the initial state (t = 0) and any time, t, in the

degradation process. Essentially it assumes that the degradation rate of drag reduction is equal to

the degradation rate of DRA:

𝐷𝑅(𝑡)

𝐷𝑅(0)=

𝑀(𝑡)

𝑀(0) (5-8)

Combining Eq. 5-8 with the degradation correlation of Eq. 5-9 leads to Eq. 5-10.

𝐷𝑅(𝑡) = 𝐷𝑅(0)𝑒−𝑘𝑡 (5-9)

𝑀(𝑡) = 𝑀(0)𝑒−𝑘𝑡 (5-10)

It can be seen that Eq. 5-10 is a solution of Eq. 5-7 if α = 1. However, it is still necessary

to determine that if α equals to 1 in the drag reduction degradation case.

Drag reduction (degradation) induced by DRAs happens when the concentration of

DRAs is very low, often in ppm (weight based). So, the interaction between different polymer

chains can be neglected (Andrade et al., 2014), and the only possible interaction in the polymer

solution is that between the solvent and the DRAs. The chemical reaction rate can be described

in Eq. 5-11.

𝑟𝑃 = 𝑘′[𝑆][𝑀] (5-11)

In Eq. 5-11, [S] represents the molecular weight of the solvent, water in this case. Since

the water structure cannot be destroyed in turbulent flow, it is reasonable to combine k’ and [S],

70

which can be rewritten as k. This indicates that the degradation of DRAs can be treated as a first-

order chemical reaction, i.e., 𝑟𝑃 = 𝑘[𝑀], or α = 1 in earlier equations.

With this theoretical analysis, the physical meaning of k and the mechanism of

degradation of DRAs can be explained in a new way: the degradation is a first-order chemical

reaction based on the molecular weight, and k is the constant of the chemical reaction rate. The

final correlation for drag reduction degradation still bears the same format as in Eq. 5-9 but

several extensions can be made based on this new theory. The results and discussions are shown

in the next section.

5.3 Results and Discussions

The degradation of DRAs is a function of time, temperature, rotation speed (shear rate),

the concentration of DRAs and molecular weight. This phenomenon can be shown in Figure 5-2.

In this figure, the torque of DI water remains as a constant over time. However, torques for the

PEO solution show a time-dependent characteristic. There is a large initial torque difference

between the dilute polymer solution and water. At this initial state, the polymer structure has

high integrity. The lower torque with the polymer solution indicates its effectiveness of flow

drag reduction. After a period of time under high shear rates, the long chain polymers can be cut

and the interaction between the solvent, water and polymer starts to change. At this point, the

increase of torque represents the degradation of drag reduction caused by degradation of the

polymers. A series of experiments with different experimental conditions are performed, and one

result is shown in Figure 5-2 because all the experimental data in other conditions have similar

trend shown in Figure 5-2.

The same time-dependent performance of the drag reduction is shown in Figure 5-3. At

the initial state, drag reduction is large because the polymer is not destroyed by the high shear

71

rate. Note that drag reduction value at the initial state, DR(0), is not measured in the

experimental setup since it is not possible to measure a torque at the initial state (t = 0). That

value can be deducted from the trend with Eq. 5-9, as did in previous studies (Deshmukh et al.,

1991). Similar to Figure 5-2, there is no need to show more results in drag reduction and

degradation since this result is similar to those from many previous studies (Lim et al., 2005 &

2007; Zhang et al., 2011). Results in Figure 5-3 further shows that the previous correlation

format shown in Eq. 5-9 can indeed provide reasonable prediction of the experimental data from

this rheometer geometry.

Figure 5-2 Torque of DI water and polymer solution (CP = 20 ppm, T = 318 K, �̇� = 8000 s-1, M =

2×106 g/mol)

Correlation development in this study will stay with the format of Eq. 5-9 but extend it to

include detailed correlation of DR(0) and k which now has a clear physical meaning. Based on

analysis in the previous section, k is the first-order chemical reaction rate. From this perspective,

72

the Arrhenius equation, Eq. 5-12, can be used for k, where k0 is the pre-exponential factor and Ea

is the activation energy of reaction.

𝑘 = 𝑘0𝑒−

𝐸𝑎𝑅𝑇 (5-12)

Figure 5-3 Experiment data and correlation by Eq. 5-9 (CP = 35 ppm, T = 318 k, �̇� = 7000 s-1, M

= 4×106 g/mol)

In the current experiments for degradation of drag reduction polymers, the degradation

rate is affected by many factors including temperature, concentration of the polymers and the

rotation speed (shear rate). To account for all these factors, the classic Arrhenius equation in Eq.

5-12 are extended by correlating the pre-exponential factor (k0) with polymer concentration (CP)

and shear rate (�̇�). The effect of temperature is already considered in the classic equation. The

correlation of k0 was done with a polynomial method in Eq. 5-13, similar to previous works

(Kalashnikov, 2002; Pereira & Soares, 2012) in predicting polymer degradation.

73

𝑘0 = 𝑎0+𝑎1�̇� + 𝑎2𝐶𝑃+𝑎3�̇�2 + 𝑎4𝐶𝑃2 + 𝑎5𝐶𝑃�̇� (5-13)

After combining the new pre-exponential factor in Eq. 5-13 and the classical Arrhenius

equation in Eq. 5-12, the format of reaction constant for the DRA degradation is shown in Eq. 5-

14.

𝑘 = (𝑎0+𝑎1�̇� + 𝑎2𝐶𝑃+𝑎3�̇�2 + 𝑎4𝐶𝑃

2 + 𝑎5𝐶𝑃�̇�)𝑒−𝐸𝑎𝑅𝑇̇

(5-14)

By correlating all the experimental data in the double gap rheometer setup in three

temperatures, 298, 318 and 338 K, the final correlation for the chemical reaction constant is

𝑘 = (5.376 × 106 + 1.027 × 104�̇� − 1556𝐶𝑃 − 822.9�̇�2 + 0.0983𝐶𝑃2

+ 6.71𝑐𝑟)𝑒−37480

𝑅𝑇̇

(5-15)

In Eq. 5-15, the activation energy, Ea is 37.48 kJ/mol. This positive activation energy

shows that increasing temperature can accelerate the degradation rate of DRAs, which is

supported by previous work (Zhang et al., 2016).

Drag reduction at the initial state, DR(0) is also affected by many factors including

polymer concentration and molecular weight, shear rate, and temperature. DR(0) is correlated

with these factors using the experimental data with a product type of correlation in Eq. 5-16:

𝐷𝑅(0) = 9.46 × 10−3𝐶𝑃

0.265 (�̇�

1000)

0.420

(𝑇

𝑇𝑡)

−2.657

(𝑀

𝑀0)

0.616

(5-16)

where T is the Kelvin temperature at experiment conditions, k; Tt is the Kelvin temperature of

transition temperature of PEO, 338 k; M0 is a reference molecular weight, 106 g/mol; 1000 s-1 is

a reference shear rate. Putting Eq. 5-16 and Eq. 5-17 into Eq. 5-9 gives the final correlation to

predict the drag reduction at any time during the degradation.

Figure 5-4 shows three sample results of DR prediction with this new correlation. An

average relative error (ARE) (Ng et al., 2002), shown in Eq. 5-17, is introduced to indicate the

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accuracy of the predictions. DRCal and DRExp represent the prediction and the experimental data,

respectively.

𝐴𝑅𝐸 =

1

𝑁∑ |

𝐷𝑅𝐶𝑎𝑙 − 𝐷𝑅𝐸𝑥𝑝

𝐷𝑅𝐸𝑥𝑝| (5-17)

These results show that the proposed correlation can predict the data reasonably well,

almost within 15% accuracy on average. This is better than some previous correlations such as

those by Bizotto & Sabadini (2005) (with ±50% accuracy) and Zhang’s et al. (2016) (whose

results showed a relatively larger error but not given directly). More importantly, this correlation

scheme provides a new mechanism of degradation as a first-order chemical reaction. All the

parameters in this new correlation have clear physical meanings. These results demonstrate the

validity of the proposed new degradation theory.

Figure 5-4 Comparison between experimental data and prediction by Eq. 5-15 and Eq. 5-16 at

different fluid and flow conditions: (1) M = 4×106 g/mol, CP = 50 ppm, T = 298 K, �̇� = 8000 s-1,

ARE = 11.7%; (2) M = 2×106 g/mol, CP = 50 ppm, T = 338 K, �̇� = 6000 s-1, ARE = 6.43%; (3) M

= 106 g/mol, CP = 35 ppm, T = 298 K, �̇� = 7000 s-1, ARE = 6.27%; (4) M = 2×106 g/mol, CP = 35

ppm, T = 318 K, �̇� = 7000 s-1, ARE = 14.8%

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Another observation from these results is that the accuracy of this correlation varies

between different experimental conditions. This indicates that there are probably other factors

that are not accounted. For example, in developing the new theory of degradation mechanism

and the new correlation, only the “overall” property of the polymer solution during degradation

is considered, i.e., the molecular weight and concentration, but the molecular weight distribution

of the polymers is neglected. In this experiment, the profile of molecular weight is not analyzed

so this profile is unknown. In drag reduction, the contribution by each molecule group (with a

molecular weight) is different, so the degradation of polymers under different molecular weight

is suspected to be different. An overall property of polymer solution can not include this

difference.

Furthermore, the fundamental assumption in Eq. 5-9, i.e., the degradation of drag

reduction is equal to the degradation of polymer may need further investigation. Although this

assumption has been applied in many previous studies, it is still based on observations, not any

rigorous chemical or physical analysis, nor is it validated by experimental results. The ratio

[(𝐷𝑅(𝑡) 𝐷𝑅(0)⁄ ] [𝑀(𝑡) 𝑀(0)⁄ ]⁄ might not equal to 1. It is suspected that this ratio is a time-

dependent parameter or one dependent on fluids/flow conditions. Based on discussions above,

further experiments are needed to investigate how the molecular weight distribution affects the

drag reduction and degradation. These issues will be addressed in the next chapter.

5.4 Summary

In Chapter 5, drag reduction and degradation by polymers in a double-gap rheometer are

investigated. A new theory based on Brostow’s assumption is proposed that the degradation of

the polymer in the turbulent drag reduction is a first-order chemical reaction. Based on this

theory and measured drag reduction data in the double-gap rotating apparatus, a correlation with

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a modified Arrhenius equation is developed to predict the drag reduction effectiveness at any

time during the degradation. The relative error of this correlation is ±15%.

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Chapter 6 A New Molecular View of Polymer Degradation in Drag-

Reducing Flow

Research presented in the earlier chapter indicate that the Brostow’s assumption might be

incorrect. In this chapter, experimental data are used to show that this important assumption is

indeed incorrect in many cases. An improved mechanism is then proposed to explain the

degradation of drag reduction. The main content of this chapter has been published as a journal

paper (Zhang, X., Duan, X., & Muzychka, Y. (2019). “Degradation of flow drag reduction with

polymer additives - a new molecular view”. Journal of Molecular Liquids, 292, 111360). The

author of this thesis is the first author of this paper. The first author analyzed the data and

prepared the manuscript. Prof. Duan and Prof. Muzychka as the second and third authors

provided their suggestions on the analysis and revisions of this paper.

6.1 Examination of Brostow’s Assumption

To examine Brostow’s assumption, one would need to measure the degradation of drag

reduction, DR(t)/DR(0) and the degradation of the molecular weight of the drag-reducing

polymer, M(t)/M(0). The former can be measured by a differential pressure sensor in pipe flow

(Zhang et al., 2018) or a torque sensor in rotational flow (Zhang et al., 2011); the latter can be

measured by a size exclusion chromatography (SEC). The degradations of drag reduction and

molecular weight have been measured in two studies, i.e., by Lee et al. (2002) in a rotational

flow and Vanapalli et al. (2005) in a pipe flow. Their data are reorganized and presented in

Figure 6-1. These data suggest that Brostow’s hypothesis in Eq. 6-1 is incorrect – the

degradation rate of drag reduction is very different than that of the molecular weight decrease.

𝐷𝑅(𝑡)

𝐷𝑅(0)=

𝑀(𝑡)

𝑀(0) (6-1)

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Figure 6-1 The relationship of DR(t)/DR(0) and M(t)/M(0): (a) revised from Lee et al. (2002) and

(b) from Vanapalli et al. (2005)

Brostow’s assumption (Eq. 6-1) provides a useful way to study the degradation of flow

drag reduction with time. It also attempts to relate the drag reduction degradation to the

(a)

(b)

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degradation of the drag reducing polymers, which gives a correct direction of finding the

mechanisms of drag reduction and its degradation from a molecular point of view. However, the

strong relationship that Eq. 6-1 has never been rigorously derived from physical or chemical

principles, nor has it been supported by experimental data. Most of the later research work only

followed the idea or format of the correlation induced by this assumption, i.e., drag reduction

degradation is a function of molecular weight decrease, but not the exact relationship that the two

rates are identical. Brostow obtained the assumption from the following statement, “At low

concentrations the changes in molecular weight can only be followed by measuring changes in

drag reduction” (Brostow, 1983; Brostow et al., 1990). The assumption was from limited

observations of experiments of drag-reducing flows, making it not suitable for all experimental

conditions, i.e., rotational flow, pipe flow, different polymer types, concentrations and flow

velocities, etc. On another important point, this assumption is based on monodisperse polymers.

While, it is well known that most drag-reducing polymers are synthetic polymers, which usually

have a wide distribution of molecular weight (Vanapalli et al., 2005). The key problem in

Brostow’s hypothesis is that it only considers the molecular weight of the free polymer but

neglects another important structure, the aggregate of polymers in the drag-reducing flow.

There are strong evidences in the literature that aggregate degradation is involved in the

drag-reducing flow. Van Dam & Wegdam (1993) investigated the degradation of drag reduction

by polyethylene oxide (PEO) with a 3.8×106 g/mol molecular weight and a 20 ppm concentration

(dilute solution). They also measured the hydrodynamic radius of the degraded polymers, RH(t),

with a dynamic light scattering (DLS) method. This radius describes the size of polymers in the

coiled state or aggregates that contain several free polymers. Unlike SEC that filters the

aggregates in the polymer solution and only measures the molecular weight of free polymers,

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DLS measures the hydrodynamic radius and it does not differentiate free polymers and

aggregates in the polymer solution. If there are no aggregates in the polymer solution, the

hydrodynamics radius represents the size of a free polymer; if both free polymers and aggregate

are in the solution, the radius of hydrodynamics provided by DLS is the average value of these

two. The hydrodynamics radius of a free polymer, RHʹ(t), is proportional to 𝑀(𝑡)𝛼 (α is a

constant between 0.5 – 0.6 for linear flexible polymers, PEO included); a free polymer with a

higher molecular weight has a larger hydrodynamic radius (Selser, 1981). Data from Van Dam &

Wegdam (1993) are analyzed and shown in Figure 6-2. The ratio of DR(t) over DR(0) is shown

as solid squares and ratio of RH(t) over RH(0) (the hydrodynamics radius at the initial state

without degradation) is shown in hollow squares. It is not surprising to see the rate of drag

reduction degradation is different than that of the hydrodynamic radiuses decrease. It is more

interesting to analyze why the hydrodynamic radius at the three early times (less than 50 s, as

shown in the doted box) remains nearly unchanged (or even slight increase). The similar

hydrodynamic radius means the average size of free polymers or aggregates is almost the same.

As shown and discussed earlier, the linear flexible PEO degrades with time in the turbulent flow.

If Brostow’s assumption were valid and there were no aggregates, then the molecular weight and

the hydrodynamic radius would continuously decrease. However, the data suggest otherwise.

The non-decreasing hydrodynamic radius is due to the aggregates in the solution. The larger

hydrodynamic radius of the newly formed aggregates makes the average hydrodynamic radius of

the degraded polymers larger. This effect disappears when the aggregates becomes much smaller

with longer residence time in the turbulent flow. Degradation of the aggregates will be discussed

in the next section.

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Figure 6-2 The relationship between drag reduction and radius of hydrodynamics from Van Dam

& Wegdam’s paper (1993)

6.2 Aggregate Degradation in Drag-Reducing Flow

Aggregates of polymers commonly exist in polymer solutions, even in a dilute polymer

solution (Peyser & Little, 1971). Shetty & Solomon (2009) used the DLS method to confirm that

aggregate structures existed in drag-reducing flows. Their results indicate that the dilute polymer

solution in the drag reducing flow is not as simple as it seems. Another evidence for the

existence of the aggregate structure is from the activation energy of the degradation process.

Dunlop & Cox (1977) showed that if the drag degradation was caused by the degradation of the

free polymer, i.e., the chain scission process, the activation energy should range from 125 to 420

kJ/mol. While, if it was caused by the aggregate decomposition, the activation energy should

range from 8 to 20 kJ/mol. A previous study (Zhang et al., 2018) showed that the activation

energy of degradation of turbulent water flow with PEO is approximately 40 kJ/mol, which is

82

between the lower and higher ranges mentioned above. This result indicates that aggregates exist

in the dilute polymer solution and the degradation of drag reduction is caused by both chain

scission and aggregate decomposition.

Formation of aggregate structures is due to the hydrophobic/hydrophilic effect between

the polymer and the solvent (Polverari & van de Ven, 1996). The common drag-reducing

polymer PEO in water are used as an example. PEO has two types of groups, CH2, a

hydrophobic group and O, OH, hydrophilic groups. PEO is a water-soluble polymer, but the

hydrophobic effect still exists due to the CH2 group. In a dilute PEO solution, the CH2 structures

from one polymer chain may attract the CH2 structures from another chain; meanwhile, OH and

O have similar behaviours. Thus, the aggregate forms in the dilute solution.

The aggregate structure is shown in Figure 6-3 as suggested by Hammouda et al. (2004).

Note that this structure is not scaled to real proportion but only an illustrative schematic. Each

colour represents one polymer chain with a given molecular weight. The dashed line between O

and OH represents the hydrogen bond, and the dashed line between CH2 and CH2 represents the

hydrophobic effect. Based on the structure of aggregate, two possible polymer degradation

mechanisms exist: (1) the decomposition of interaction between polymers, i.e., only the

hydrogen bond and hydrophobic effect are destroyed but the polymer chain remains integral, as

shown in Figure 6-3a; or (2) the decomposition of polymer chain, i.e., the chain scission happens

at the polymer chain of aggregate, as shown in Figure 6-3b.

83

(a)

(b)

Figure 6-3 Possible aggregate degradation mechanism in drag-reducing flow

Additional experimental studies were conducted to investigate the behaviors of polymer

aggregates during turbulent flow drag reduction. The details of the experimental setup and

materials can be found in my previous paper (Zhang et al., 2018). Figure 6-4 shows two types of

degradation tests of PEO in water: (1) continuous degradation, where polymers experience 6-min

degradation continuously; (2) the batch degradation, where polymers experience a 2-min

degradation, followed by a 10 min break (without flow movement); then more cycles of the

degradation are repeated until the polymers experience a 6-min degradation test in total.

The results show that the two types of degradation tests give almost the same results

(with only 2% offset due to the minor difference of the initial torque at the beginning of the

tests). These results indicate that a standing time does not help the polymer aggregate recovery to

regain the drag reducing ability. This phenomenon happens because the behavior of polymer

aggregates in drag-reducing flows follows the mechanism shown in Figure 6-3b. If the

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mechanism was the one shown in Figure 6-3a, the standing time should help the polymer

aggregates to reform (Polverari & van de Ven, 1996; Duval & Gross, 2013) and their drag

reduction ability should recover. However, it does not happen. Thus, the mechanism in Figure 6-

3b is the correct one to describe the degradation of aggregates in the drag-reducing flow. Here it

is not intended to show that the interaction between two polymer chains cannot be interrupted by

the turbulent flow. In fact, this interaction can be interrupted but can reform due to the recovery

after degradation.

Figure 6-4 Continuous and batch degradation test of PEO (molecular weight 4 106 g/mol,

concentration 20 ppm, temperature 20 °C and shear rate 6500 s-1)

Note that the polymer degradation behavior in the drag reducing flow is different than the

behavior of surfactants. For surfactants, the recombination of surfactant micelle can recover the

drag-reducing ability of surfactants. The high shear rate can only cause the degradation of the

85

micelle, not the surfactant itself, so the drag-reducing ability can recover after degradation (Liu

et al., 2018).

6.3 Molecular Weight Distribution Shift in Drag-Reducing Flow

The other issue in the degradation of drag reduction is the molecular weight distribution

after the degradation. The polymer chain is cut into small pieces after experiencing a high shear

rate in the turbulent flow. So, the molecular weight should decrease, but it is still not clear if the

molecular weight distribution after degradation is broader or narrower or remains the same. In

dilute polymer solutions, the degradation of polymers is a random process, but it is possible that

the chain scission happens in the middle of the chain, which is supported by Bueche (1960) from

theory, and Horn & Merrill (1984) in experiments with Polystyrene and Choi et al. (2002) & Lim

et al. (2005) in experiments with DNA. Buchholz et al. (2004) further proved that polymers with

a higher molecular weight are more likely to degrade than that with a lower molecular weight.

One example is used to manifest the shift of molecular weight distribution after the

degradation in a drag-reducing flow. Assume that one drag-reducing polymer has an average

molecular weight of M, with a distribution profile, M/2, M and 2M, shown in Figure 6-5. At a

given time in the drag-reducing flow, polymers with a molecular weight M may experience the

chain scission once, and the molecular weight becomes M/2; polymers with a molecular weight

2M may experience the chain scission twice due to the higher molecular weight, and their

molecular weight also becomes M/2; polymers with a molecular weight M/2 may not experience

the chain scission at all due to its low molecular weight, and their molecular weight remains the

same, M/2. After this degradation process, the average molecular weight becomes M/2, therefore

the molecular weight distribution becomes much narrower. Some other studies showed different

results. For example, Liberatore et al. (2004) reported that the average molecular weight and

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distribution remained the same after degradation of drag reduction in the flow. This is due to the

high polymer concentration (approximately 1000 ppm in these studies) that is much higher than

the dilute solution mentioned above. In high concentration (semi-dilute) solutions, aggregates are

more likely to form and affect the drag reduction, so less free polymers are involved in the drag

reduction and degradation. In this case, the free polymers could remain integral. Before the SEC

test, the aggregates are filtered, and the integral polymers after the degradation are injected into

the SEC, so the measured molecular weight and its profile remain the same as in the original

condition.

Figure 6-5 The molecular weight distribution before and after degradation

Besides what is shown above, another drag reduction test can validate our view of the

distribution shift, as shown in Figure 6-6. It shows that the profile indeed becomes narrower after

more passes (longer degradation time).

87

Figure 6-6 Molecular weight distribution of degraded polymer in drag reduction (data from

Vanapalli et al. (2005))

6.4 Summary

In Chapter 6, the degradation of flow drag reduction and the degradation of the drag-

reducing polymers are investigated. The critical assumption of Brostow that the rate of drag

reduction degradation is identical to the rate of molecular weight decrease of the drag-reducing

polymer is shown to be not correct in rotational flows and pipe flows. This is mainly due to the

existence of polymer aggregates. Experimental results show that the molecular weight of the

degraded polymer in the dilute solution becomes lower and the molecular weight distribution

becomes narrower. Finally, the degradation mechanism of polymer aggregates in the drag-

reducing flow is proposed: the turbulent flow causes the chain scission of the aggregate and the

degraded aggregate losses its drag-reducing ability.

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Chapter 7 Mechanism of Drag Reduction and Degradation from

Chemical Thermodynamics and Kinetics

In previous chapters, it is seen that drag reduction is related to the polymer relaxation

process and degradation is related to the chain scission. In this chapter, these two major

conclusions are combined, and a new explanation of drag reduction by polymers from the

chemical thermodynamics and kinetics is proposed. The main content of this chapter has been

published as a journal paper (Zhang, X., Duan, X., & Muzychka, Y. (2020). “Drag reduction by

linear flexible polymers and its degradation in turbulent flow: a phenomenological explanation

from chemical thermodynamics and kinetics”. Physics of Fluids, 32(1), 013101). I am the first

author of this paper. I analyzed the data, prepared the draft paper, and made the revisions. Prof.

Duan and Prof. Muzychka as the second and third authors provided their suggestions on the

analysis and revisions of this paper.

7.1 Explanation of Drag Reduction by Polymers

The polymers’ behavior in the flow can be treated as a conformational phase change

process between the coiled state and the stretched state (Fidalgo et al., 2017), further regarded as

a chemical reaction (Ferguson et al., 1987), so the chemical thermodynamics and kinetics are

used to analyze this process.

Figure 7-1 shows the schematic of this transition, i.e., polymer configuration changes

from the coiled state to the stretched state in a dilute solution of certain concentration in flow

under a constant temperature. Here, this transition can be treated as a reversible process within a

given time, which is supported by experimental results (Fidalgo et al., 2017) shown in Figure 7-

1. In this experiment, a special polymer, i.e., DNA, is used. It has a uniform molecular weight

distribution. In Figure 7-1 x/L is the ratio of the real-time length of DNA to the maximum length

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measured by a fluorescence microscope. Figure 7-1 shows two processes, i.e., relaxation (from

the stretched state to coiled state) and stretching (from the coiled state to stretched state, also

regarded as elongation). The time-dependent length can be approximated by a cosine function, as

shown by the solid curve in Figure 7-1. Besides, the order of magnitude of the length ratio of

stretched-state polymer over the coiled-state polymer is 10, validated by Virk (1976). This

periodic function indicates that our view of the polymer configuration is correct: a transition

between the coiled and stretched states exists and this transition is reversible within a limited

time. The limited time is emphasized since degradation happens – to be discussed later.

Figure 7-1 Polymer transition between the coiled state and the stretched state (data from Fidalgo

et al. (2017))

Next, the equilibrium constant K of this reversible transition process is defined below. In

traditional chemical thermodynamics, the reactant and product concentrations at the equilibrium

90

state are involved in the equilibrium constant. In this case, the exact number of polymers in the

coiled and stretched states should be employed, however, current measurement technology

cannot achieve that. To propose a phenomenological explanation, the microscopic feature of the

polymers in the coiled and stretched state are used, ΔPS (pressure drop when polymers are in the

coiled state, equal to the one of pure solvent, which will be explained below) and ΔPP (pressure

drop when polymers are in the stretched state) under the same flow rate. Here the viscosity

difference of pure solvent and polymer solutions is not considered since polymers at ppm level

have negligible influence on the viscosity (Zhang et al., 2019).

𝐾 =

∆𝑃𝑃

∆𝑃𝑆 (7-1)

Usually, the pressure drops with and without polymers are compared to identify if the

drag reduction happens. Slightly different from this approach, the pressure drops with polymers

in coiled states and the one with polymers in stretched states are compared. Here it is assumed

that the pressure drop in the pure solvent without polymers is equal to the pressure drop when the

polymer is in the coiled state, and further, the pressure drop when polymers in the coiled state

and stretched state are compared to determine if the drag reduction occurs. This assumption is

based on the fact that the polymer in the coiled state can be regarded as a small particle (Virk,

1976) and it has no drag reduction ability in a low concentration: many independent works found

that the minimum concentration of micro or nano-particles to reduce the flow friction is several

hundred ppm (Pirih & Swanson, 1972; Pouranfard et al., 2014; Radin et al., 1975), much higher

than the polymer concentration of the dilute solution, usually less than 100 ppm. Thus, it is

assumed that polymers in the coiled state in the dilute solution cannot induce drag reduction.

The Gibbs free energy of this reaction ΔG is shown in Eq. 7-2. ΔGS and ΔGP represent

the Gibbs free energy of the polymers in the coiled state and stretched state in the turbulent flow.

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The physical meaning of these terms will be explained below. R is the ideal gas constant, 8.314

J⋅mol−1⋅K−1; T is the Kelvin temperature (K). When the drag reduction happens, ΔPS is larger

than ΔPP and the equilibrium constant (K) is less than 1.

∆𝐺 = ∆𝐺𝑃 − ∆𝐺𝑆 = −𝑅𝑇 ln 𝐾 (7-2)

From Eq. 7-2, it can be seen that the ΔG of this reaction is greater than 0 when the drag

reduction happens, so this reaction (drag reduction) is non-spontaneous at a given concentration.

In drag reduction by polymers, an onset point exists at a given concentration (Virk, 1975). The

drag reduction phenomenon only happens in turbulent flow (based on the Reynolds number,

4000 for pipe flow) even though there are some counterexamples (Choueiri et al., 2018). At a

given concentration, only a high enough velocity that can induce the stretching of the polymer

chain can cause drag reduction. This onset point explains the physical meaning of the critical

energy in Camail et al.’s work (2009).

The Gibbs free energy of polymers in the stretched state is higher than that of the

polymer in the coiled state from Eq. 7-2. This higher potential means that the stretched polymer

has a higher chemical potential thermodynamically, so the stretched polymer has higher reactive

activity. This higher activity causes the stretched polymers to interact with turbulent structures,

and the energy in the vortex is reused in the stream-wise direction (laminarization). Finally, drag

reduction happens, and the purely viscous turbulent flow (when no polymers are added) is

laminarized and becomes elongational turbulent flow (Zhang et al., 2018 & 2019). Here the

physical meaning of the Gibbs energy of polymer is clear: it represents the energy stored in the

polymer. A longer polymer chain means a high Gibbs energy which can interact with the

turbulent flow structures, and this interaction leads to a better drag reduction. This explains why

a higher molecular weight has a better effect on drag reduction (Zhang et al., 2019).

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Turbulent flow provides an unstable environment for polymers in the coiled state

thermodynamically so that free polymers can be stretched, and the drag reduction happens in this

elongational turbulent flow. However, this elongational turbulent flow also causes the polymers

to become unstable kinetically. Because the Gibbs free energy of the stretched polymer is higher

than the one in the coiled state, it has more potential to interact with the turbulent structure. In

turn, the turbulent structure also interacts with the free polymer, which means that the polymers

experience the chain scission, also known as the degradation (Zhang et al., 2018). Recent work

showed that polymer degradation truly happened in both rotational and pipe flows (Zhang et al.,

2019). Molecular weight measured by size exclusion chromatograph decreased with time,

indicating that the chain scission happens.

The free polymer behavior in drag reduction are discussed. While in the polymer

solution, another structure coexists with the free polymer - the polymer aggregate, which

contains several polymer chains. The behavior of polymer aggregates is similar to free polymers,

which can interact with turbulent structure (Kalashnikov & Kudin, 1973) and be degraded by the

chain scission, detailed information provided in ref. by Zhang et al. (2019).

In summary, an explanation of the drag reduction by polymers is proposed in Figure 7-2,

and this explanation is divided into three stages.

(1) Dissolution. The solid polymer is dissolved in the solvent, and two types of polymer form

in the solution, free (single) polymers and the polymer aggregates.

(2) Unstable thermodynamics for polymer chain stretching. When the turbulent flow

overcomes the onset point to stretch the free polymer and polymer aggregate, the drag

reduction happens due to the unstable thermodynamic environment. This process is

reversible.

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(3) Unstable kinetics for polymer chain scission. After the free polymer and polymer

aggregates experience long-time turbulent flow, the chain scission happens, and it causes

the degradation of drag reduction due to the unstable kinetic environment. This process is

irreversible.

Figure 7-2 The proposed explanation of the drag reduction and degradation by linear flexible

polymers

7.2 Discussions

One advantage of this theory is that it can explain why polymers cannot reduce the

friction completely - friction not being equal to zero. Theoretically, if there were enough

polymers in a turbulent flow and all the polymer elasticity potential was used to damp the

turbulent structures, the flow friction could be eliminated entirely. But this never happens, which

is due to our proposed explanation from the Gibbs free energy view. The Gibbs free energy

describes the “maximum non-expansion reversible” work in a thermodynamic system under

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constant temperature. Please note that maximum work only happens in a reversible process, not

for an irreversible process. As discussed above, both reversible (chain stretch) and irreversible

(chain scission) processes are involved in drag reduction by polymers. Thus, the whole drag

reduction process is an irreversible process. Therefore, the friction in turbulent flow cannot be

reduced to 0.

In the proposed explanation, the polymer chain has two features, coiled-stretched

transition and chain scission, the former being a periodic process (reversible) and the latter one a

non-periodic process (irreversible). Overall, it can be treated as a damped oscillation process

(damped for chain scission and oscillation for coiled-stretched transition).

The statement above about damped oscillation is the behaviour of drag-reducing polymer

in turbulent flow from the microscopic view, which is also reflected in the drag reduction from a

macroscopic view. Bewersdorff & Petersmann (1987) and McComb & Rabie (1978) measured

the drag reduction in different locations of a straight pipeline independently: in Figure 7-3a (from

ref. by McComb & Rabie, 1978) and 7-3b (from ref. Bewersdorff & Petersmann, 1987), x is the

length from the polymer injector to the measuring point, m. There is a “quasi-cyclic” (introduced

by McComb & Rabie (1978) originally) drag reduction at different locations in Figure 3a: near

the polymer injector, the drag reduction is small due to the entrance length effect; at far-enough

locations from the polymer injector, the drag reduction oscillation happens. The quasi-cyclic

behavior of drag reduction further supports our explanation from a molecular behaviour view: in

the coiled-stretched transition, the length of the polymer in the coiled state is short, whose

extreme condition can be regarded as a small particle and in general can be regarded as a

polymer with a low molecular weight, so the drag reduction is low; the length of the polymer in

the stretched state is long, which can be regarded as a polymer with a high molecular weight, so

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the drag reduction is high. Besides, the transition of the polymer between the coiled-stretched

states from the microscopic view is reflected in the drag reduction oscillation from the

macroscopic view.

Does the oscillation truly happen or is it from the uncertainty of the measurement?

McComb & Rabie (1978) repeated the same experiment, also shown in Figure 7-3a. The results

show that data repeatability is good at least for the data in the marked box, which are measured

at far-enough locations from the polymer solution injector. These two almost-identical

experimental results with repeatable trends indicate that the oscillation truly happens.

Bewersdorff & Petersmann’s results (1987) in Figure 7-3b further validate the existence of this

drag reduction oscillation.

However, it seems that there is no degradation of drag reduction in these results, which

do not agree with the proposed explanation. This is due to the short distance from the polymer

injector to the last measuring point: in Figure 7-3a, x/d is 200 approximately, which means that

the distance of measured drag reduction data is very short in a lab-scale experiment. Besides, the

Reynolds number in these experiments is 45,000, indicating a high velocity. Short distance and

high velocity (short residence time) show that the polymer residence time in the drag-reducing

pipe flow is small (in Figure 7-3b, the maximum residence time in the flow is 3.5 s) so that the

degradation is not obvious. If the residence time is long enough, the degradation can be

observed. Camail et al. (1998) tested the polyacrylamide in a rotational flow for more than three

days and the results are shown in Figure 3c. In this test, a peak of the drag reduction appeared,

which can be viewed as the drag reduction fluctuation (oscillation); after that the degradation of

drag reduction started to happen until it almost disappeared in 2.5 days. More data from oil

transportation pipeline support the ideas. The article by Strelnikova & Yushchenko (2019)

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provides many drag reduction data from field tests with polymers in several oil & gas

transpiration systems. Figure 7-3d shows one of those tests where the damped oscillation trend of

drag reduction by polymers versus distance.

Figure 7-3 Drag reduction oscillation versus distance or time (a: data from McComb & Rabie

(1978); b: data from Bewersdorff & Petersmann (1987); c: data from Camail et al. (1998); d:

data from Strelnikova & Yushchenko (2019))

More data from ref. by Strelnikova & Yushchenko (2019) are shown in Figures 7-4a and

7-4c. The damped ossification feature of the drag reduction along the pipeline distance is similar

to a wave. This inspires us to use the Fourier series, a method for wave study, to analyze the

characteristic of the drag reduction versus distance. In Fourier series, a wave can be separated by

a constant and the sum of several simple harmonic waves with different frequencies. Strictly

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speaking, Fourier series is for the periodic function, not for non-periodic function such as the

damped oscillation of the drag reduction. But this method is still extended here since it is helpful

to explain the contribution of drag reduction by the molecular weight distribution.

The Fourier series is shown in Eq. 7-3.

𝐷𝑅(𝑥) = 𝑎0 + ∑ 𝑎𝑘cos(𝑘𝜔𝑥) + 𝑏𝑘sin(𝑘𝜔𝑥)

𝑛

𝑘=1

(7-3)

In Eq. 7-3, ak and bk are fitting parameters. Eq. 7-3 can be converted to Eq. 7-4 by

combining sine and cosine terms.

𝐷𝑅(𝑥) = 𝑎0 + ∑ 𝑐𝑘cos(𝑘𝜔𝑥 − 𝜑𝑘)

𝑛

𝑘=1

(7-4)

ck (amplitude) and φk (phase angle) are defined in Eq. 7-5 and Eq. 7-6. From Eq. 7-5, ck

(amplitude) represents the ability of drag reduction, a higher ck means a higher drag reduction

potential. From Figure 7-1, the polymer (with a given molecular weight) relaxation process can

be correlated by a cosine fitting curve and this relaxation process induces the drag reduction as

discussed above. So, in Eq. 7-4, each cosine term indicates one polymer with a specific

molecular weight.

𝑐𝑘 = √𝑎𝑘

2 + 𝑏𝑘2 (7-5)

𝜑𝑘 = tan−1

𝑏𝑘

𝑎𝑘 (7-6)

As an example, n is set as 5 in this analysis, indicating that there are 5 types of polymer

with different molecular weights. In Figure 7-4b and 7-4d, it is seen that the drag reduction can

be predicted by Eq. 7-4, which verifies that our analysis method with the Fourier series is useful.

The solid bar, shaded bar and hollow bar from Figure 7-4b and 7-4d represent the absolute value

98

of an/a0, bn/a0 and cn/a0. The results show that neither an nor bn is regular, but cn is: cn decreases

with the increasing n, which means that the drag reduction ability is large when the frequency is

low. This happens because most synthetic drag-reducing polymers have a wide molecular weight

distribution (Zhang et al., 2019). This distribution contains high and low molecular weights. The

polymer with a high molecular weight has a long relaxation time (a low frequency, also meaning

a small n) and a large contribution to the drag reduction.

Figure 7-4 Field drag reduction data in an industrial pipeline and their results after Fourier series

transformation (data from Ref. by Strelnikova & Yushchenko (2019))

7.3 Summary

In Chapter 7, a new mechanism of the drag reduction and degradation from chemical

thermodynamics and kinetics is proposed: the drag reduction phenomenon by linear flexible

99

polymers can be explained as a non-spontaneous irreversible flow-induced conformational-

phase-change process that incorporates both free polymers and aggregates. The entire non-

equilibrium process is due to the chain scission of polymers. Drag reduction results from a

macroscopic view and polymer behaviours from microscopic views further support this theory.

A Fourier series expression indicates that the molecular weight distribution can affect drag

reduction.

100

Chapter 8 Conclusions and Future Work

8.1 Conclusions

In this thesis, flow drag reduction by polymers and its degradation are investigated

analytically and experimentally. The main contributions and conclusions from this work are

summarized below.

A new model for drag reduction by polymers is proposed and validated by previous

experimental data. Based on the FENE-P model, the semi-analytical solution assumes complete

laminarization in the turbulent flow and can predict the upper limit of drag reduction in the pipe

flow. This model is also used to explain the optimum concentration and velocity at the maximum

drag reduction.

A semi-empirical correlation is developed for the estimation of the drag reduction in

turbulent pipe flow in dilute polymer solutions, based on the Weissenberg number and polymer

concentration. The relaxation time in Weissenberg number is based on Zimm’s theory, so the

problem that the relaxation time cannot be measured in a low concentration is solved. The

definition of low concentration is also clarified. Experimental data from the flow loop are used to

develop the correlation, and the relative error between experimental data and predicted values is

within +/- 30%. This correlation is further validated by previous experimental data in laboratory

and industrial flow systems.

A new theory is proposed that the degradation of polymer in a turbulent drag-reducing

flow is a first-order chemical reaction based on the famous Brostow’s assumption. According to

this theory, a correlation with a modified Arrhenius equation is developed using data from a

rotational flow to predict the drag reduction during the degradation with a ±15% relative error.

101

The Brostow’s assumption is shown to be incorrect after it is checked with experimental

data from in rotational flows and pipe flows. It is found that the main reason is that this

assumption neglects polymer aggregate in the drag-reducing flows. The molecular weight of the

polymer decreases, and the molecular weight distribution becomes narrower. An improved

mechanism of aggregates in drag-reducing flow is proposed: the polymer aggregate degradation

happens due to the chain scission, and degraded aggregates have no drag reduction ability.

Chemical thermodynamics and kinetics are introduced to develop a new theory of the

drag reduction and degradation mechanism. Under this frame, the drag reduction phenomenon by

linear flexible polymers can be explained as a flow-induced non-spontaneous irreversible (chain

scission) conformational-phase-change process (relaxation), which involves both free polymers

and aggregates. This is supported by drag reduction data from the macroscopic view and

polymer performances from the microscopic tests.

8.2 Future Work

Based on the research work completed in this thesis, the following future research topics

seem promising.

8.2.1 Environmental-Friendly Drag-Reducing Agents

The research of environmental-friendly drag-reducing agents was not new. Hoyt and Soli

(1965) started to investigate this topic for the first time. There are several different

environmental-friendly drag-reducing agents: polymers from algal (Gasljevic et al., 2008; Hoyt,

1970; Hoyt & Soli, 1965; Ramus et al., 1985), from bacteria (Abdulbari et al., 2018; Kenis,

1968; Sar & Rosenberg, 1989) and from plant (Soares et al., 2019). Polymers from these natural

resources are considered as non-toxic to the environment which will not cause environmental

problems. These agents could reduce the environmental problem introduced by the synthesis

102

polymers being used nowadays. More research efforts should be spent on the polymers from

natural resources.

8.2.2 Anti-Degradation

The other issue about the synthetic drag-reducing polymer is degradation. Future work

can target to improve the anti-degradation property of drag-reducing polymers. It is well known

that most drag-reducing polymers cannot resist the chain scission by turbulent flow, thus

degradation always happens. Several groups have already started to investigate how to

synthesize better long chain flexible polymers to resist the chain scission (Camail et al., 2003;

Gryte et al., 1980; Ma et al., 2003; Nifant'ev et al., 2018). In these papers, authors show the

comparison between the commercially available drag-reducing polymers and synthetic polymers,

indicating that many emerging synthetic polymers have better anti-degradation ability. However,

few synthetic drag-reducing polymers with degradation resistance ability are available

commercially, so more future work in this area is desired.

8.2.3 Oil-Soluble Polymers and Multiphase Flow

Currently, most research on drag reduction by polymers was performed in single-phase

water flow, therefore, the research results only have limited industrial applications such as in the

oil-gas industry. Thus, more oil-soluble polymers should be involved in the research. Besides,

drag reduction in oil flow is still a single-phase flow research, which is not very helpful in the oil

and gas industry where multiphase flows are often seen. There are many studies of drag

reduction in single-phase flow, with more than 1000 journal papers. While the number of drag

reduction studies on multiphase flow is much smaller, with only 70 journal papers published

until this thesis is finished. Future researches should focus on the drag reduction by polymers in

multiphase flows - a more promising area.

103

8.2.4 Synergy of Drag Reduction by Polymers and Surface Modification

The synergy effect of drag reduction by polymers and special surfaces should be further

studied. Until now, there are limited studies that combine these two strategies of flow drag

reduction. If these two methods are combined, the drag reduction could be higher than the one by

each method. But the synergy effect is not clear. Mizunuma et al. (1999) mentioned that the V-

shape riblet could increase the drag so this surface itself was not a drag-reducing surface, but

when drag-reducing polymers were added in the turbulent flow, the drag reduction occurred.

Similar observations were reported by Koury & Virk (1995) and Huang & Wei (2017). In

another work by Huang et al. (2016), the combined drag reduction by grooved structure and

surfactant was shown to be greater than the drag reduction by each method, which is also

supported by Abdulbari et al. (2018) and Christodoulou et al. (1991). Anderson et al. (1993)

suggested that drag-reducing polymers did not affect the drag reduction induced by a specific-

designed surface, meaning that the surface had the drag reduction ability and polymers had no

effect on the drag reduction by the surface. In this situation, more work should be completed to

investigate the combination of these two methods and make it clear that which combination of

surface (with a particular structure) and polymer can lead to a positive synergy effect - a better

drag reduction performance.

As mentioned earlier, drag-reducing polymers can also reduce the corrosion to the

pipeline inner surface. If the drag reduction by polymers and a certain anti-corrosion surface with

drag reduction ability can be combined, the total friction will be reduced significantly and the

pipeline can be maintained for a long time due to the anti-corrosion effects by polymers and

surfaces.

104

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Appendix

Appendix 1 How to Obtain the Weissenberg Number from 𝝁𝑺, u, d, and N

TABLE A1 Detailed procedure of combining these four variables as one dimensionless number,

the Weissenberg number, based on Zimm’s theory (1956)

Step Parameter

/Group

Unit Parameter

/Group

Unit Parameter

/Group

Unit Parameter

/Group

Unit

1 𝜇𝑆 kg ⋅ m-

1 ⋅ s-1

u m ⋅

s-1

d m N ...

2 𝑁1.8𝜇𝑆 kg ⋅ m-

1 ⋅ s-1

u m ⋅

s-1

d m ... ...

3 𝑎3

𝑘𝐵𝑇𝑁1.8𝜇𝑆

s u m ⋅

s-1

d m ... ...

4 (𝑎𝑁0.6)3

𝑘𝐵𝑇𝜇𝑆

s u m ⋅

s-1

d m ... ...

5 (𝑎𝑁0.6)3


s 8u m ⋅

s-1

d m ... ...

6 (𝑎𝑁0.6)3


s 8𝑢

𝑑

s-1 ... ... ... ...

7 (𝑎𝑁0.6)3


8𝑢

𝑑

...

NOTES: 1: Four remaining parameters are listed; 2: The degree of polymerization, N, and

solvent viscosity, 𝜇𝑆, are combined. The unit of the new group 𝑁1.8𝜇𝑆 has the same unit as 𝜇𝑆; 3:

The monomer length of the PEO, a, Boltzmann constant, kB, and temperature, T, are constants in

each experiment. Therefore, these constants, a3/kBT and 𝑁1.8𝜇𝑆 are combined; 4: This new

group, 𝑎3𝑁1.8𝜇𝑆 𝑘𝐵𝑇⁄ , is rearranged to (𝑎𝑁0.6)3𝜇𝑆 𝑘𝐵𝑇⁄ ; the latter one is equal to the relaxation

134

time from Equation (2); 5: A constant 8 is multiplied by u; 6: To eliminate the unit of length, 8u

is divided by d. Therefore, a new group 8u/d forms, which is equal to the shear rate at the wall in

Equation (4); and 7: (𝑎𝑁0.6)3𝜇𝑆 𝑘𝐵𝑇⁄ and 8𝑢 𝑑⁄ are combined to have the Weissenberg number.

135

Appendix 2. Why the Correlation Format of Eq. 4-8 Is Proposed

In addition to the main reasons related to previous experimental and analytical results

explained in the paper, the format can be further explained from a statistic point of view. If one

assumes that CP and Wi follow a second order polynomial (without a constant, since DR should

be 0 when no polymers are added) in Equation (A1), the P values for CP, CP2, Wi, Wi2 and CPWi

are 1.47 × 10-6, 0.86, 0.62, 0.07, and 0.68, respectively. A lower P value (usually < 0.05) means

that a variable is more relevant to the equation. From these five P values, it can be seen that CP is

the most relevant to Eq. A1 since its P value is the lowest. CP2, Wi, and CPWi are not relevant

since their P values are much greater than 0.05. The P value of Wi2 is 0.07, slightly greater than

0.05, which means that this variable has a high potential of relevance to the drag reduction

correlation if a proper correlation format is used, Eq. 4-8 as an example. If Eq. 4-8 format is

used, the P values for CP and Wi2 are 3.05 × 10-27 and 1.09 × 10-10, meaning CP and Wi2 are

relevant to the correlation. In this condition, it is believe that Wi2 not Wi should be chosen for the

correlation, as shown in Eq. 4-8:

𝐷𝑅 = 𝐴1𝐶𝑃 + 𝐴2𝐶𝑃2 + 𝐴3𝑊𝑖 + 𝐴4𝑊𝑖2 + 𝐴5𝐶𝑃𝑊𝑖 (A1)

136

Appendix 3. List of Publications

Published Journal Papers

Zhang, X., Duan, X., & Muzychka, Y. (2018). Analytical upper limit of drag reduction

with polymer additives in turbulent pipe flow. Journal of Fluids Engineering, 140(5),

051204. https://doi.org/10.1115/1.4038757

Zhang, X., Duan, X., & Muzychka, Y. (2018). New mechanism and correlation for

degradation of drag-reducing agents in turbulent flow with measured data from a double-

gap rheometer. Colloid and Polymer Science, 296(4), 829-834.

https://doi.org/10.1007/s00396-018-4300-4

Zhang, X., Duan, X., & Muzychka, Y. (2019). Degradation of flow drag reduction with

polymer additives—A new molecular view. Journal of Molecular Liquids, 292, 111360.

https://doi.org/10.1016/j.molliq.2019.111360

Zhang, X., Duan, X., & Muzychka, Y. (2020). Drag reduction by linear flexible polymers

and its degradation in turbulent flow: A phenomenological explanation from chemical

thermodynamics and kinetics. Physics of Fluids, 32(1), 013101.

https://doi.org/10.1063/1.5132284

Zhang, X., Duan, X., Muzychka, Y., & Wang, Z. (2020). Experimental correlation for

pipe flow drag reduction using relaxation time of linear flexible polymers in a dilute

solution. The Canadian Journal of Chemical Engineering, 98(3), 792-803.

https://doi.org/10.1002/cjce.23649

Journal Papers under Preparation

Zhang, X., Duan, X., & Muzychka, Y. Mini-review of drag reduction and degradation by

polymers: history, application, research and theory.

137

Published Conference Papers

Zhang, X., Duan, X., Muzychka, Y. (2017, May). Analytical solution of drag reduction

by chemical additives in turbulent pipe flow. In 26th Canadian Congress of Applied

Mechanics. Canadian Congress of Applied Mechanics.

Zhang, X., Duan, X., Muzychka, Y., & Wang, Z. (2018, November). Predicting drag

reduction in turbulent pipe flow with relaxation time of polymer additives. In 2018 12th

International Pipeline Conference. American Society of Mechanical Engineers.

Posters and Presentations

Zhang, X., Duan, X., Muzychka, Y., & Wang, Z. (2017, May). A possible mechanism for

degradation of drag-reducing agents in turbulent flow in rheometers. Memorial

University of Newfoundland and Canadian Society for Mechanical Engineering

Symposium.

Zhang, X., Duan, X., Muzychka, Y. (2018, May). Degradation of drag-reducing polymers

from chemical dynamics view. Memorial University of Newfoundland Engineering

Research Day.

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